This is a pretty straight forward question that deserves a straight forward answer.

In the books of the canonized New Testament, I can find no place where Paul ever directly quotes any sayings of Jesus that are recorded from any other source than himself.

On several occasions, Paul attributes to Jesus what he claims to have heard in his visions (see the above answers) but he never reaffirms any of the direct teachings of Jesus as recorded in any other source.

Is it true that Paul did not quote Jesus a single time in the entire New Testament

Yes, this is true, and the best source for this claim is Paul's own description of where he got his information. It does matter, though, how you define "quote Jesus."

The pithy sayings and moral teachings attributed to Jesus are preserved mainly in Matthew and Luke, which were not written down until roughly 20 or 30 years after Paul's death. Mark probably also postdates Paul's death. We don't know how Jesus's sayings were being preserved during Paul's lifetime, maybe orally and later in some written form, but it's not at all surprising that they weren't transmitted to Paul using the technology of the day, which was letters -- Paul was the one who started the whole epistle thing in early Christianity.

And Paul himself tells us that this is not the case -- he says that he received his knowledge of Jesus's teachings and messianic identity not through someone sending him a letter, but by divine revelation. There are many cases in the Pauline epistles where, to the modern mind, it would make sense to quote Jesus's words in order to support the point Paul is making. So for example, Paul discusses dietary laws, but he doesn't quote Mark 7:15. But Mark hasn't been written down yet, and if Paul were to say, "Someone told me that Jesus said, 'There is nothing from outside of the man that going into him can defile him,'" it wouldn't be particularly authoritative to people in the ancient world. The Hebrew bible and divine revelation both provide direct access to God's authority, and those are the sources of authority that Paul uses.

Of course if you broaden the definition of "quote Jesus" enough, you can certainly find examples. A huge percentage of what Jesus is quoted as saying in the gospels is either a quote from the Hebrew bible or a paraphrase of it. If Paul writes, "Jesus says, ..." and it's a quote from Isaiah, then you could call that quoting Jesus, or you could just say that Paul had heard that Jesus used this verse from Isaiah for this purpose, and Paul is quoting Isaiah.

And if you believe that the resurrected Jesus revealed himself directly to Paul, then of course you have examples like 2 Corinthians 12:9.

Does Paul ever actually quote Jesus anywhere?

Acts 26:14-18

And when we were all fallen to the earth, I heard a voice speaking unto me, and saying in the Hebrew tongue, Saul, Saul, why persecutest thou me? it is hard for thee to kick against the pricks. 15 And I said, Who art thou, Lord? And he said, I am Jesus whom thou persecutest. 16 But rise, and stand upon thy feet: for I have appeared unto thee for this purpose, to make thee a minister and a witness both of these things which thou hast seen, and of those things in the which I will appear unto thee; 17 Delivering thee from the people, and from the Gentiles, unto whom now I send thee, 18 To open their eyes, and to turn them from darkness to light, and from the power of Satan unto God, that they may receive forgiveness of sins, and inheritance among them which are sanctified by faith that is in me.

In 1 Timothy 5:18 Paul quotes both the Torah and Jesus at the same time.

For the Scripture says, "Do not muzzle the ox while it is treading out the grain." (Deuteronomy 25:4) and "the worker deserves his wages." (Luke 10:7)

You can see that Paul is quoting Jesus exactly, word-for-word in Greek from Luke 10:7.

Stay there, eating and drinking whatever they give you, for the worker deserves his wages. Do not move around from house to house.

It is obvious that Paul was not present with the other Apostles and so it would be impossible for him to "remember" what Jesus had said, however this does not preclude direct revelation from Jesus (which is what his other quotes are). But if the question is trying to cast aspersion on Paul's "fitness" or authority to write Scripture then Peter even lays this doubt to rest when he lumps Paul's writings in "with the other Scriptures" in 2 Peter 3:15-16.

I don't know if you intend to place restrictions on what constitutes "quoting Jesus" (compared to the Gospels or Acts), but these instances are at least Paul attributing commands or words to him.

1 Corinthians 7:10-11 (NASB)
But to the married I give instructions, not I, but the Lord, that the wife should not leave her husband (but if she does leave, she must remain unmarried, or else be reconciled to her husband ), and that the husband should not divorce his wife.

2 Corinthians 12:9 (NASB)
And He has said to me, "My grace is sufficient for you, for power is perfected in weakness." Most gladly, therefore, I will rather boast about my weaknesses, so that the power of Christ may dwell in me.

Acts describes an instance where Paul, saying his final farewell to the Ephesians, quotes Jesus.

Acts 20:35 (NASB)
"In everything I showed you that by working hard in this manner you must help the weak and remember the words of the Lord Jesus, that He Himself said, 'It is more blessed to give than to receive.'"

Paul spent more time (in writing) quoting the Old Testament than he did quoting anyone who was his contemporary.

It might not be entirely reasonable to make an issue of this if we are only considering his writings. There are no historical narratives in the canon that are attributed to Paul, and that's where Jesus is, by far, the most quoted. You can probably find as many quotes or attributions to Jesus in Paul's writings as in any other New Testament epistle in the Canon.

Words of Jesus' Ministry

In Acts 20:35, Luke quotes Paul as quoting Jesus, but the original quote is not attested in any of the four gospels. (Though that doesn't mean Jesus didn't say it.)

By all these things, I have shown you that by working in this way we must help the weak, and remember the words of the Lord Jesus that he himself said, 'It is more blessed to give than to receive.'

In 1 Corinthians 7:10-11 Paul refers to "the Lord" (which is what he usually called Jesus) giving certain instructions on divorce. They clearly correspond to words Jesus spoke in Matthew 19:8-9 and Mark 10:11-12:

To the married I give this command—not I, but the Lord—a wife should not divorce a husband 11 (but if she does, let her remain unmarried, or be reconciled to her husband), and a husband should not divorce his wife.

In 1 Corinthians 11:23-25, Paul quotes Jesus' words at the Last Supper, which are also recorded in Matthew 26:26-28, Mark 14:22-24, and Luke 22:19-20:

For I received from the Lord what I also passed on to you, that the Lord Jesus on the night in which he was betrayed took bread, and after he had given thanks he broke it and said, "This is my body, which is for you. Do this in remembrance of me." In the same way, he also took the cup after supper, saying, "This cup is the new covenant in my blood. Do this, every time you drink it, in remembrance of me."

In 1 Timothy 5:18, Paul quotes "Scripture," but the only other place in the Bible that says it is Luke 10:7, where Jesus is sending out the Seventy.

For the scripture says, "Do not muzzle an ox while it is treading out the grain," [Dt 25:4] and, "The worker deserves his pay." [Lk 10:7]

Jesus' words directly to Paul

• In Acts 9, Jesus spoke to Paul in a vision. Paul is recorded telling others about it in Acts 22 and 26

• In 2 Corinthians 12:9, Paul writes that Jesus said to him, "My grace is enough for you, for my power is made perfect in weakness."

1 Corinthians 11: 23-26
For I received from the Lord what I also handed on to you, that the Lord Jesus, on the night he was handed over, took bread, and, after he had given thanks, broke it and said, “This is my body that is for you. Do this in remembrance of me.” In the same way also the cup, after supper, saying, “This cup is the new covenant in my blood. Do this, as often as you drink it, in remembrance of me.” For as often as you eat this bread and drink the cup, you proclaim the death of the Lord until he comes.