Dumbledore destroys the first Horcrux with Godric Gryffindor's sword, but, as revealed in "Harry Potter and the Deathly Hallows," he also recognizes the stone in the ring as the Resurrection Stone, one of the three Deathly Hallows.
Dumbledore destroys the first Horcrux with Godric Gryffindor's sword, though he is mortally injured by the ring's curses after putting the ring on his finger. Dumbledore had learned the stone is, in fact, the Resurrection Stone, one of the three Deathly Hallows.