Sauron needs the Ring to be assured that others do not have it. It was his greatest weakness.
This is, of course, for two reasons:
- His enemies could use the Ring to bolster their own military might (he thinks this likely after Aragorn uses the Palantir)
- The Ring could be destroyed (Sauron did not think anyone would have the audacity to do this)
This thread goes into some great detail about what was going through the mind of Sauron when the Ring was destroyed.
The actual passage that user @Valorum posted is below.
And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him, and his Eye piercing all shadows looked across the plain to the door that he had made; and the magnitude of his own folly was revealed to him in a blinding flash, and all the devices of his enemies were at last laid bare. Then his wrath blazed in consuming flame, but his fear rose like a vast black smoke to choke him. For he knew his deadly peril and the thread upon which his doom now hung. From all his policies and webs of fear and treachery, from all his stratagems and wars his mind shook free; and throughout his realm a tremor ran, his slaves quailed, and his armies halted, and his captains suddenly steerless, bereft of will, wavered and despaired. For they were forgotten. The whole mind and purpose of the Power that wielded them was now bent with overwhelming force upon the Mountain
[The Return of the King: Being the Third Part of the Lord of the Rings][1]