With the Ring, Sauron would undoubtably defeat Gondor and Rohan. The armies of the Free Peoples would be no more, except to fight an underground war against his armies. The remaining Dwarves would bolt their doors tight, as they did in the second age, and the elves would either flee Middle Earth or would go into hiding. Hobbits, too, would be enslaved if they did not go into hiding.
However, Sauron's dominion would not last forever.
As Arda will reach its true potential as Iluvatar imagined it by the End, at some stage Sauron would have to be defeated by someone in order for this to happen. If his power became too great for the free peoples to defeat alone, at some stage I'd imagine the Valar would interceed as they did against Morgoth.