Are there famous battles that took place on ice?

I have not been able to find any reliable source for this, but at least a source that mentions a primary source:

In the siege of Stockholm during the conflict between King Albert of Sweden and Queen Margaret of Denmark in 1393/94, several ships from Albert's side were stuck in the frozen Baltic Sea and built palisades covered with ice in order to (successfully) defend themselves from the Danes:

Acht dieser Schiffe froren vor der schwedischen Küste im Eis ein. Da ihr Hauptmann einen dänischen überfall befürchtete, ließ er im Wald Holz schlagen und mit den Stämmen einen Wall um die Schiffe errichten. Dieser Wall wurde mit Wasser übergossen, so dass das Eis die Stämme fest zusammenhielt und nach außen eine gläserne Oberfläche bildete. Als die Dänen angriffen, konnten sie trotz ihrer übermacht den glatten Wall nicht überwinden.

(from http://www.kriegsreisende.de/renaissance/vitalier.htm)

Not a battle, but at least a military operation was the hunt across the Kurisches Haff

The Vistula lagoon (just southwest from the Kurisches Haff) was the site of aerial attacks during the evacuation of East Prussia in early 1945:

From January until March 1945 throughout the Evacuation of East Prussia, refugees from East Prussia crossed the frozen lagoon on their way to the west after the Red Army had reached the coast of the lagoon near Elbing on January 26. Attacked by Soviet fighter aircraft thousands of them were killed or broke through the ice.

(from https://en.wikipedia.org/wiki/Vistula_Lagoon)

andejons already mentioned the March across the Belts, but another example is the final battle of the Dacke War (Dackefejden in Swedish) 1543 took place on the ice of the lakes outside Virserum. Nils Dacke's troops were mainly farmers and they were successful when waging a guerilla war in the forests, but when they met the German landsknecht mercenaries in open terrain they were easily defeated.

Another one that is quite interesting is the capture of the Dutch fleet at Den Helder that is said to be the only case when a cavalry beat a navy.

There was an incident during the Battle of Austerlitz where the French bombarded the Russians while they were retreating across the Satschan frozen ponds. This incident caused the ice to break up and hundreds of Russians drowned, perhaps up to 2,000 and they also lost several artillery pieces.

This event was not critical to the battle however as the Russians had already lost and were withdrawing. There is also some contention that the story has been made up or at least exagerated by Napoleon himself and the reality is very few Russian bodies were found after the lakes were drained.

http://ehr.oxfordjournals.org/content/XVII/LXVII/537.extract

https://en.wikipedia.org/wiki/Battle_of_Austerlitz

During the Soviet-Finnish "Winter War" of 1939/40 the Soviets attempted to outflank the Finns many times over frozen lakes and the Gulf of Finland. Well-supplied Finnish coastal artillery would blast great holes in the ice, swallowing the exposed Soviet infantry.

[February 10th & 11th] the Russians tried for the first time to outflank the [Mannerheim] line by sending powerful infantry columns on a long, curving march across the thick ice. Before these forces could come ashore behind Finnish lines, however, they were spotted and taken under fire by the coastal batteries in the Koivisto sector, particularly the six-, eight-, and ten-inch weapons emplaced near Saarenpää, on Koivisto Island, and at Humaljoki on the mainland...

The six- and eight-inch weapons were supplied with shrapnel shells that were fused to detonate in airbursts over the heads of the Russian columns like gigantic shotgun blasts. The heavy coastal battery, armed only with shells designed to pierce a battleship's armored deck, could not cause that kind of damage with their elephantine projectiles, but the enormous weight and velocity with which they struck tore great holes in the surface ice, so that each successive strike enlarged the fractures until they turned into chasms. Hundreds of men drowned in the cold black waters, sucked down by the weight of their gear, or froze to death in a matter of seconds as they tried to swim to solid ice.

Source: "A Frozen Hell" by William R. Trotter p 219

The Finns would use frozen lakes as makeshift forward airbases. Here's a Bristol Blenheim light bomber on frozen lake Jukajärvi.

Source: Talvisodan pikkujättiläinen (1999) p. 341, orginally from SA-kuva (Finnish Army Pictures)

One example is the Battle of Bogesund, a part of the internal conflicts of the Kalmar Union, which led to then Swedish regent Sten Sture the younger being mortally wounded after a cannonball had bounced on the ice, and ultimately allowed Kristian II of Denmark to seize the Swedish throne.

Other battles has taken place on Storsjön, Bysjön, and Viken. There is also a mythological batttle on Vänern.

For a really history-changing event (even if was not an actual battle) which was made possible due to winter conditions, there is the March across the Belts, during which king Charlex X of Sweden managed to cross into the Danish island Sjaelland and threaten Copenhagen. The peace that followed transferred large parts of present Sweden from Denmark-Norway.

While not directly a scene for major combat, the Lake Ladoga ice road was vitally important to Leningrad during WW2.

Oh boy, this is the moment to speak of the most awesome battle ever.

You see, in winter 1794, a French Hussard regiment was sent to prevent a Dutch fleet, stuck in Den Helder, from rejoining British forces. The Dutch Republic was in a state close to civil war and the fealty of those ship was in question.

And so, a cavalry regiment had the exceptional opportunity to charge across the ice and capture a 14 ship navy formation on horse back !

But ... as always, reality is boring, there probably was no fight, and the Dutch admiral most likely just swore an oath not to sail against France while everyone drank hot milk.

I'll be in my office, drinking, listening to classics and crying over our lost opportunities.