Isildur does not destroy the ring by casting it into the fires of Mount Doom. Instead, he keeps it for himself.
But Isildur and his party are ambushed by remaining orcs. Isildur puts on the ring, which makes him invisible, and tries to escape but doesn't make it. Isildur is killed and the ring slips off his neck.
The ring lies at the bottom of some random lake for two and a half thousand years until it becomes merely a legend and is forgotten.