Remember, also, that the ring is simply a sidebar in this prequel. If a reader were to finish The Hobbit without ever reading the LOTR trilogy, that reader probably wouldn't think twice about the ring at all, it would just be one magical item encountered by Bilbo on this adventure. Why Tolkien chose to keep this novel's focus away from the ring when the rest of the series is strictly about the ring is up to debate. At the end of The Hobbit, however, we see Bilbo more concerned about getting his hobbit hole and all of his possessions back than he is about the ring or even the small chests of gold and silver. We really don't learn about the outcome of the ring until the beginning of Fellowship when it is passed down to Frodo through Gandalf.

Bilbo keeps the ring at the end of the story and presumably hides it to keep it safe.