Clueless Sauron?

According to Gandalf in the chapter "The Shadow of the Past", Sauron had believed that the One Ring "had perished; that the Elves had destroyed it". Yet surely Sauron must have well aware that the One Ring still existed somewhere in the world.

After all, had the Ring been destroyed, Sauron's shadow would not have been able to take shape and grow again in Mirkwood, and the Nazgul would have been obliterated. At what point then did Sauron become aware that the One Ring was still in existence?