**Potential Spoiler Warning**
Well, I'm happy I got so many replies to the question. There are just too many posts to do quotes from, so I'll try to address the arguments as I come to them.
It seems I am in agreement with nessonite. The ring had indeed corrupted Frodo at the end but I don't think Frodo was truly aware of the ring's power. He most likely would have turned out like Gollum, but I'm sure the Nazgul would have got him almost instantly after he put on the ring if Gollum wasn't there. Sam I'm sure would have tried to help Frodo, but I think he was beyond even that. I don't think Frodo would have killed Sam though... Smeagol, when he killed Deagol, showed that he didn't have much of an ethic to begin with, but Frodo throughout the books/movies showed loyalty to Sam.. even with the ring getting "heavy" on him. Frodo probobly would have just walked by Sam.
As to the ring "being" Sauron, that's not quite true. Sauron, when he had the 19 rings forged, created the One Ring completely on his own, and vested his own power into it in order to control the other 19. So while in a sense the ring is Sauron, it is only his power in it, not his actual spirit. He however does have a link with it though, and while it still exists, he can maintain a physical form (the Eye). He can also maintain some power over the Nazgul (who were severely weakened when Sauron was defeated -- they were far, far stronger in the Second Age, especially the Witch King).
But you are right in that he expected the ring to return to him one way or the other. Idealy it would be the Nazgul to return it, but he figured anyone with the ring would challenge him. His reasoning for this is the fact that he figured that anyone who had the ring knew of its power. This is one of the main reasons Gandalf sent Frodo. Gandalf (as he said) knew he would not be able to withstand the corruption. Frodo had a better chance because he knew he himself knew he had an item of power, but didn't know its power (other than going invisible).
Someone with knowledge of the ring could challenge Sauron himself and become the next Dark Lord. This would effectively destroy Sauron as much as if it were thrown into Mt. Doom. Of course the person who challenged (and won) would be corrupted, but they would *not* be Sauron. They would have Sauron's Maia power, but without his influence. They would also control the Nazgul and Barad-dur - and the orcs would quickly follow the new Lord of the Ring, whoever it may be. As was said by Alchemy, and it is said in the books as well, Sauron doesn't think they would give the ring to a Hobbit to take to Mt. Doom to destroy it. He did however suspect it. I'm sure he had to at least do that. If it wasn't for Aragorn using the Palantir before heading to Gondor (this is book version), I'm sure Sauron would have gotten the ring. Sauron assumed Aragon had the ring, and thus his thought that the ring was already in Mordor (or coming to Mordor) was gone.
"They are definitely no Sauron of Mordor. He was/is a great necromancer and had untold powers that the peoples of Middle Earth never really fully grasped or could gauge until they actually fought him, which they did. Remember, he was only narrowly defeated."
Not quite true. The elves definately knew of his power. So did the 3 wizards mentioned in the book. I'm sure anybody who was anybody knew of Sauron's power, as they most likely knew of his origins. Sauron was lieutenant to Melkor (who dwarfs Sauron in power -- even with the Ring) in the First Age of Middle-Earth. All the elves in Middle-Earth are aware of this since they were either in Middle-Earth to begin with (the Sindar - Legolas is one) or followed Feanor to help reclaim the Silmarils (the Noldar - Galadriel is one). They also fought in the wars against both Melkor in the First Age, and Sauron in the Second. They knew Sauron at the height of his power.
Oh, and Saruman definately wanted the ring to become the next Dark Lord. He would not have returned the ring to Sauron. This is why Sauron was so afraid when Merry looked into the Palantir. Sauron thought he had the Ring, which meant in turn that Saruman had the ring. Sauron knew Saruman would betray him if he ever got the ring and thus sent the Nazgul asap to retrieve it. By the time the Nazgul got there though, the Ents had already destroyed the area surrounding Orthanc. It was then that Sauron also found out that infact, Saruman did not have the ring and assumed the hobbit escaped to Gondor with Gandalf, who was in turn going to give it to Aragorn.
Heh, whoa, I sure deviated from the question. Well, long story short, I as I said earlier, I don't believe Frodo had the power or knowledge to challenge Sauron, and the Ring would have been reclaimed had Gollum not been there.
What after that? Well, Sauron would have reformed, regained his power, renewed the Nazgul and trounce what was left of resistance on Middle-Earth. The elves would have sailed away and Sauron would rule Middle-Earth. For how long.. I don't know. Probobly until the Last Battle... which was never really finished by Tolkien. The gist of it is Melkor returns from Exile and renews Sauron and the fallen Balrogs and recreates all his old armies of dragons and orcs and then a huge Battle Royale ensues. Of couse, none of this is official, only speculation.