The Longetivity of Hobbits.

[Inziladun]

Originally Posted by Faramir Jones

This was to ask the question that if something like the Ring makes a mortal being from a particular race live far longer than his or her normal time span, is that person still a member of that race?

The keeper of the Ring is not fundamentally altered in what they are. A Man remains a Man, a Hobbit (or quasi-Hobbit), a Hobbit. He "borrows" Sauron's natural immortality while he bears it, and when that influence is removed, nature takes its course.
Witness how Bilbo was affected. At the time of the Council of Elrond, Bilbo was old, but still sound in mind and body. After the Ring was destroyed, he very quickly showed his age by sleeping a great deal and becoming very forgetful.