As Victim of Circumstance makes clear in his answer, the fact that the One Ring could be detroyed at Mount Doom was known to the elves at the end of the second age, and this information probably came from Celebrimbor. However, at the Council of Elrond, Elrond says
Gandalf has revealed to us that we cannot detroy it by any craft that we here possess.
So, the elves did not know that the ring could only be destroyed at Mount Doom. Only Gandalf seems to have known any more about the destruction of Rings of Power, and he knew a lot. Aside from pointing out that the Elves could not destroy the One (and that dragons destroyed four of the Seven), Gandalf rules out several other possibilites:
... he [Tom Bombadil] cannot alter the Ring itself.
(The Council of Elrond)
Not even the anvils and furnaces of the Dwarves could do that.
(The Shadow of the Past)
... nor was there ever any dragon, not even Ancalagon the Black, who could have harmed the One Ring, the Ruling Ring, for that was made by Sauron himself.
(The Shadow of the Past)
In fact, Gandalf states unequivocally that
There is only one way: to find the Cracks of Doom in the depths of Orodruin, the fire-mountain, and cast the Ring in there.
(The Shadow of the Past).
I think it's important to stress that Ancalagon (had he been alive) could not destroy the Ring because it was 'made by Sauron himself'. My own reading of this is that Gandalf knew destroying the ring without returning it to Mount Doom would require a power greater than Sauron's. How else could he have known that no other method would work, when none had been tried? However, as Gandalf himself puts it:
I knew much and I have learned much. But I am not going to give an account of all my doings to you.