[The Might]

Letter #246 was just what I was going to mention after reading through the thread.
However the outcome would be clear - if Frodo had taken the Ring Sauron would have eventually come himself to Mount Doom and would have claimed the Ring
Also, Tolkien explores another possibilty. He says that if Gollum had indeed repented for his deeds, as it seemed earlier near Cirith Ungol, he might have taken the Ring from Frodo and would have cast himself in Mount Doom willfully in order to save Frodo, thus also defeating Sauron.

If Frodo had claimed the Ring somewhere else in ME I am not so sure what the outcome would have been. It could be that the wise would have been able to intervene in time and take the Ring away, but you never know. If he would be able to keep it I guess he wouldn't have great plans, he never really had dreams of glory and power. Probably he would end up similar to Gollum.