Two major things that are missing in this discussion:

1. Omani Empire, and the arab slave trade in the Indian Ocean.
2. Portuguese slave trade starts already in the 1300 in Africa, with sea and land expeditions from Morocco to the south (guinea).

1519 date is clearly wrong. 1815 date is anglo-centric, but fine.

Is it true that slavery was endemic in Sub-Saharan Africa previous to the establishment of the trans-Atlantic slave trade? Yes, was that answered, it most certainly was and somewhat sophisticated. Research African empires that preceded the Europeans, such as the Dahomey and Ashanti empires and their early slave trade. A bigger question may be has any distinguished race or origin emerged without slavery? The earliest tribes invading another's land, the victor would conquer and kill, but also absorb some, maybe children who could still be molded, maybe wives forced into their tribe to replace lost wives, or just to build the size of the tribe. This was slavery in its earliest form, although I bet class was not inherent then and a slave or their offspring could probably have risen in that tribe. Comanche Indians were a good example of a nomadic society who took in slaves, but if the slave proved their courage or value, they were accepted into the tribe. The last great Comanche chief, Quanah Parker, was a young white captive used as a slave, who eventually proved his worth to be their leader. Looking at the earliest conflicts between societies, a battle only had three results for the losing side: killed/tortured then killed; escape/retreat; or slavery (even if given nice accommodations or a high ranking spouse, it was still a lifestyle by force, slavery). It occurred everywhere, the Aztecs, Mayan and Incas, Mesopotamia, Early China and India, rest assured it was well established in Africa and well beyond Egypt, prior to the arrival of the mid century Europeans or early Americans.

The answer to this question isn't a yes or a no, but i would like to make some important points.

1) Slavery wasn't considered a big crime during the ancient times, and it did exist across every part of the world in different forms, more or less severely.

2) Using the term "Sub-Saharan Africa" is overly broad, as there various nations with different value system, across the geographical region referred to as "Sub-Saharan Africa" hence drawing a conclusion as to what was endemic to this entire region (Sub-Saharan Africa) could be absolutely miscalculated.

Slavery was the norm in the world since time immemorial. Listening to many, they think it a Caucasian/African American issue only. A simply amazing fact is that slavery was all but eliminated in Christendom by the year 1,000 even in barbaric northern Europe. This has much to do with Christian leaders who were former slaves including St. Patrick as well as Queen Bathilde, wife of Clovis II who forbade slavery in the Merovingian kingdom in the 700's. If you think of what was going on in the Middle East, China, India, Africa, as well as amongst native Americans this is simply remarkable. Thomas Aquinas condemned slavery. Now, sadly it was reintroduced by Moorish slavers in Southern Europe just before the discovery of the New World. One of the major slave trade centers was Genoa, from which Columbus hailed. Note that Columbus wanted to enslave the indigenous of the new world, this was rejected outright by Queen Isabella of Spain as she considered them her subjects. She did allow the importation of slaves from Africa.

The rest, they say is history of slavery in the west. Slavery, of course, continued unabated in the non-Christian Old world. But then, it was the collective efforts of Christian abolitionists in Europe such as William Wilberforce, John Wesley and Charles Spurgeon and the abolitionists in America such as William Lloyd Garrison: "Christian charity while it begins at home goes abroad in search of misery...the condition of slaves, in a religious point of view is deplorable..." It took 400 years to finally rid the new world of slavery. But even then, it was still a revolutionary step if one looks at what was going on in the middle east, India, Africa and China. And it was the culturally insensitive, intolerant British that outlawed slavery in India and much of the Arab world.

Why did abolition happen in the Christian world? One can find examples of purported Christians using scripture to justify the institution. Certainly slavery was a reality in both old and new testament times. One can look at old testament injunctions on the care of bond-servants and say that "justifies" slavery. Paul asks the runaway slave Onesimus to voluntarily go back to his master but then asks his Christian master to release Onesimus. Does this "justify" the abhorrent institution? Quite the contrary. Lincoln cited the scripture in Genesis, "by the sweat of your brow (and not someone else's) will you eat your bread". The American abolitionists cited the Golden rule, Matthew 7:12. Paul wrote, "In Christ Jesus, there is neither slave nor free" Galatians 3:28.

Slavery did not disappear from Europe. It was common practice in the Muslim states up to the end of the Reconsquista.

Portugal practiced slavery too. It was far less common, but they still did it. Portugal and Slavery

They never stopped it before the Atlantic Slave Trade they started. The Portuguese outposts in West Africa were almost "on the road" to current Brazil.

The Muslim states did have an influence on this point. But they can bear no responsibility for our own doing.

Yes, there was slavery in Black Africa. Shockingly, blacks are human... with all human qualities and defects. Who would have imagined?

All human societies past say neolithic have slavery. Black Africa has a huge diversity of civilizations, so it had slavery. It's a really obvious pattern. The only reason for people NOT seeing this obvious pattern is if they consider blacks as "something different" - hence the flippant answer above.

Even shorter form: Duh, yes. Like everybody else at that time, dude.

I am writing this answer mainly to provide some sources.

As for the claim "1519 to 1815 is the period of Christian slave trading", the first date is glaringly wrong.

The Siete Partidas, a Spanish law code from the thirteenth century, gives rules for slavery at Partida 4, Title 22, showing that slavery then existed. (1807 Real Academia de Historia edition consulted on Hathi Trust.)

In Peter Russell's biography of Prince Henry "the Navigator" (Dom Henrique of Portugal), we see that Europeans were buying slaves in West Africa in the early to mid fifteenth century, as Europeans became capable of navigating to West Africa. They also enslaved the population of the Canary Islands.

Many biographies of Columbus describe his sending captives to Spain to be sold as slaves in the 1490s. For example, in Bartolomé de Las Casas' history first published in 1875 and available in such places as Hathi Trust, this is found at vol. 2, pp. 322ff.

The second date is also obviously wrong, since slave trading within the slaveowning countries continued until they abandoned slavery, until 1865 in America for example.

As regards the broader question of whether "slavery was reintroduced to Europe in the 16th century, following . . . its expansion in the Muslim world", leaving aside the error in the date, I can't for sure disprove it. However, for a yes answer, one would need to show that slavery disappeared in Europe sometime in the Middle Ages. There was a process whereby slaves became serfs, but I have no good references for this. (An old publicly available book is Paul Allard, Les Origines du Servage en France (1913), but its reliability and applicability outside France is unclear.) Whether this process went so far that there were no slaves in Europe seems difficult to know absent something like legislative evidence of abolition. One would also need to show that slavery "expanded" in the Muslim world. It seems difficult to prove this (and I can't read Arabic), unless one is saying that the growth of the Muslim world by the conquests of the seventh and eighth centuries CE itself implies an "expansion" of slavery there.

True slavery was still common in Western Europe until early medieval times.

Most peasants were serfs and were not at the time considered slaves. However whilst serfdom and slavery have some theoretical and legal differences, the conditions of serfdom fulfill all the criteria of slavery as it is recognised today. It is important to recognise those distinctions for historical accuracy, but equally it is important to use modern standards to recognise how little practical difference those distinctions made to people trapped in that situation.

It is also important to note that these differences only apply to adult males. Wives and children were frequently considered chattel property of the male head of the household. Wives could be sold, and Oliver Twist famously covers the sale of children as apprentices in Victorian England.

Slavery was common to most ancient societies. On a more specific question about Muslim slave trade in Africa, it lasted from the 7th to the 19th centuries, and some estimates claim 18.5 million victims (counting only dead, not those traded).

Of course all numbers of this sort can be only very rough estimates.

Source: Matthew White, The great big book of horrible things, Northon and Co, NY, London, 2011.

According to this book, the total numbers of victims of trans-Atlantic slave trade and East African slave trade are approximately equal, but of course the trans-Atlantic trade lasted much less time.

EDIT. Let me add some detail from this book. The author says that Middle Eastern slave trade has been studied much less than the transatlantic one. So he uses the same dead/transported ratio that is used for the trans-Atlantic trade, and this ratio is 3/2. (For every 3 reaching the destination 2 die).

When counting Muslim slave trade, the author does not include those who were captured by the Muslim traders (mentioned in other answers) and sold to be transported to the West. These are included under trans-Atlantic trade.

He mentions those Western Europeans caught and sold by the Muslims (0.5 million by his count). But he never mentions slaves exported by the Muslims from Eastern Europe (Slavs). East European slave trade (mostly through Crimea and Balcans) was also a large enterprise which lasted many centuries. Some authors even speculate that the word "slave" comes from the word Slav.

I am going to do something apparently silly, and answer my own question.

That's because the question is not originally mine, but by a user named Maria BI. She asked this in Skeptics stackexchange, and people there voted to migrate it here (which seems reasonable; it is a question about History, after all).

But then the question was closed here (and so much closed that I couldn't even comment or find the discussion about its closure). My impression is that this was a knee-jerk reaction, and that a huge part of such reaction is due to Maria BI being a newbie. Such impression is reinforced by the fact that I essentially reasked the question, and it seems to be well accepted now.

I was hoping that I would not have to answer it, as other people would give adequate answers. But I am not satisfied by the answers so far, which generally suffer from two problems: 1. the identification of any kind of forced labour with slavery - which may be morally satisfactory, but is historically wrong - and 2. the consequent idea that "slavery was the rule after the invention of agriculture and before the industrial revolution".

Those ideas are false. In part, they stem from the fact that most ancient historiography we have access to is either Greek or Roman; Rome and Greece were societies based on slavery, and their writers tended to see "slavery" in other social formations. This is not unusual - Portuguese chroniclers wrote about Tupi-Guarani "kings", even though the latter never had anything similar to a monarchy; and modern writers often talk of "capital" in obviously pre-capitalist societies.

In fact, slavery was rather the exception than the rule in human history (which is not to say that forced labour was rare). For instance, Perry Anderson writes (in Passages from Antiquity to Feudalism, and citing Finley, Between Slavery and Freedom):

The originality of this mode of production must be underlined. Slavery itself had existed in various forms throughout Near Eastern Antiquity (as it was later to do elsewhere in Asia): but it had always been one juridically impure condition - frequently taking the form of debt bondage or penal labour - among other mixed types of servitude, forming merely a very low category in an amorphous continuum of dependence and unfreedom that stretched well up the social scale above it.

Most agricultural, pre-industrial societies relied in the exploitation of peasants through rent, ie, the concession of land against the payment of dues either in labour, or as a part of agricultural production, or in money. We have discussed it here.

As far as I am informed, actually only the Greek and Roman civilisations were extensively based on slavery (the kingdom of Palmira was the most probable exception). Slavery did exist in other civilisations, but was for the most part exceptional, and usually restricted to personal services, not being the main form of labour in agriculture. Again, I refer to Anderson:

it was a residual phenomenon that existed on the edges of the main rural work force. The Sumerian, Babylonian, Assyrian and Egyptian Empires - riverine states built on intensive, irrigated agriculture that contrasted with the light, dry-soil farming of the later Mediterranean world - were not slave economies, and their legal systems lacked any sharply separate conception of chattel property. It was the Greek city-states that first rendered slavery absolute in form and dominant in extent, thereby transforming it from an ancillary facility into a systematic mode of production.

Even in Greece and Rome, the full development of slavery was only possible due to the reforms that abolished debt-slavery. As Anderson says,

it was not fortuitous that the salvation of the independent peasantry and the cancellation of debt bondage were promptly followed by a novel and steep increase in the use of slave-labour, both in the towns and countryside of classical Greece. For once the extremes of social polarization were blocked within the Hellenic communities, recourse to slave imports was logical to solve labour shortages for the dominant class.

Now to the substance:

There was some kind of "slavery" in Subsaharian Africa before European colonialism set foot there. It completely differed in kind, substance, and scope, from slavery as we know it in the "new world" - United States, Cuba, Brazil. For instance, the Portcities website for "Bristol and Transatlantic Slavery" explains this as

Slavery existed in Africa, but it was not the same type of slavery that the Europeans introduced. The European form was called chattel slavery. A chattel slave is a piece of property, with no rights. Slavery within Africa was different. A slave might be enslaved in order to pay off a debt or pay for a crime. Slaves in Africa lost the protection of their family and their place in society through enslavement. But eventually they or their children might become part of their master’s family and become free. This was unlike chattel slavery, in which enslaved Africans were slaves for life, as were their children and grandchildren.

It was - of course - not linked with racialism, it was usually restricted to personal services, particularly to the local nobilities, and was subjected to little trade, if at all. Slaves were essentially war booty. Noblemen probably make them gifts for friends or foes, but the practice of making slaves for the express purpose of selling them only took off once European colonialism entered the equation. There was little slave labour in agriculture or husbandry; the women would serve in the harems of conquering lords, both men and women would be domestic servants in the great houses.

So yes, European slavetraders certainly belaboured upon an existing tradition of enslaving conquered people; but to them "slavery" meant something completely different than to African local warlords - quite probably, something completely unimaginable from the perspective of that people.

Islam certainly does not forbid slavery; it merely restricts it to non-believers. What Islam does forbid is racial discrimination, so slavery in the Muslism world had little to do with race. Muslism warriors and merchants would enslave "Blacks", as in subsaharian Africans; but they would also enslave "whites" like Serbians or Bulgarians. They did establish routes for slave trade, but those usually had to do with the East, via Indian Ocean, not with Europe or, much less, the new world (so, the Europeans did not inherit such routes, but created new ones). Slavery in Muslism domains also remained basically an issue of services, not of agriculture, and the status of slaves in such places was very different from their status in the West (to the point that the ruling layer of Egypt, at some point, was actually composed of people who were technically slaves; see, for instance, The Mamluks in Egyptian Politics and Society, edited by Thomas Philipp and Ulrich Haarman).

For the sheer scale of trans-Atlantic slave trade vis-a-vis the Middle Eastern (trans-Saharian, Red Sea, trans-Indic), Nathan Nunn (scholar.harvard.edu/nunn/files/hup_africa_slave_trade10.pdf‌​) puts figures at 12 million traficked Africans in the trans-Atlantic, within a total of 18 million in all routes from the 15th to the 19th century; so the average annual impact of the Atlantic trade would twice as important as that of all Arab-dominated routes; but notice that figures for the trans-Atlantic trade sharply increased in time, so the brunt of the manhunt was in the century between 1750 and 1850, when more than 7,500,000 slaves were trafficked, and it became so significative and devastating that even the sex ratio in West Africa was disrupted (see the Transatlantic Slave Trade Database.)

Now a bit of ad hominem: Stefan Molyneaux, is spite of having formal academic training as a historian, is what would be best called a "crank": someone who spouses "non-mainstream" (to say the least) views on history, for the end of both political propaganda and personal self-promotion. So whatever he writes or youtubes should be taken with a huge amount of salt - probably enough to boost Utah's economy. Regarding his political stances, they seem to be served by a kind of revisionism that denies the excepcionality of European colonialism, and tries to "blame" slavery on Muslims - if not on Blacks themselves.

Islam dominated slave trade between the 7th and the 15th century, while the Christians entered the market of human flesh much later - 1519 to 1815 is the period of Christian slave trading.

It is incorrect to say Christians were not involved in the slave trade before or after this era. He appears to be referring to the Atlantic Slave Trade, and conflating this with slavery generally. It is also incorrect to imply earlier slavery was the preserve of Islam. There was never a clean separation between Christian and Muslim involvement.

It is true that slavery was an important part of Islamic civilisation, from the Barbary Slave Trade, to the Janissaries. But the period he describes, which is roughly speaking the medieval era, was inclusive of slavery conducted by non-Muslim Europeans. It's also worth remembering that these markets were often intertwined; Muslims, Christians, Jews, pagans, were all involved.

The Viking Slave Trade continued until the 11th century. The largest slave markets in Europe were Dublin and Constantinople. Slavery was common in Byzantium long after Constantine began converting the empire to Christianity.

The targeted slavery of Africans in Spain began in the 15th century. Though slaves had been owned by Muslims, Christians, Jews, in Spain for a long time. This leaves one wondering why he dates Christian/European slave trading 1519-1815.

The elephant in the room of course is that slavery was abolished in Russia in 1723. As this answer explains, the reality was that life for many serfs in Russia was little better than it had been for slaves prior to their emancipation. So slavery continued in essence until Alexander II emancipated the serfs in 1861.

Serfdom was not limited to Russia, and existed in much of Europe until relatively recently. Conditions for serfs, especially unfree serfs, were essentially slavery. The distinction between a peasant, serf, and slave varied from place to place. Often the distinction was in name only.

Africa

Slavery is an ancient universal institution, which appeared independently in all cultures and societies which reached a certain level of productivity per capita.

Early hunter-gatherers did not have it because each tribe member could barely sustain himself, so there was no incentive for slavery (i.e., male captives were murdered, often as sacrifices, while females bore children for their captors), but agriculture provided ample opportunities to exploit forced labor (and thus both male and female captives usually survived).

Thus slavery in Sub-Saharan Africa appeared at least 5000 years ago.

The statement that "there has never been slavery in an agricultural society X" is akin to statement "there exists a flat (as opposed to spherical) planet Y" - the burden of proof is on the author of such an implausible statement, not on someone who rejects it.

(Western) Europe

Medieval European economy was based on a feudal rather than slave system: peasants were (usually, more or less, especially during and after the Plague) personally free, but most land was owned by the nobility. There was no place for major slave labor use.

This does not mean that slavery as an institution was unknown. Crusaders brought back black and Moorish slaves as a curio or a personal attendant, but these were irrelevant to the economic structure of the society.

During the Age of Exploration and American conquest, slavery became economical again, and that's when it made its way back into the European economy: cash crops can be more efficiently produced by large plantations employing slave labor (or, today, using machinery) than by small family farms like in feudal Europe.

Europeans found it more economical to buy Black slaves from the Muslim traders in Africa than to enslave the locals.

PS1. I am fully aware of the cruelties of serfdom, but I am also aware that it was significantly different from slavery (specifically, serfs owned property and worked at least some time - in fact, most of the time in the most advanced societies - for themselves, while modern slaves never had any economic standing). You also need to take the answer in the context of the question, which specifically inquires about trade in Black African slaves.

PS2. Of course, ancient Greece and Rome were major slave-driven powers. However, that was about 1,000 years before the period in question. By the time Atlantic slave trade came about, Slavery was replaced with Serfdom in Europe.

PS3 See also What was the impetus for some African nations to participate in the slave trade?