The Longetivity of Hobbits.

[Inziladun]

Originally Posted by Faramir Jones

Is there any reference to a wearer of the Ring '"borrowing" Sauron's natural immortality while he wears it' in LotR? I might be mistaken, but I only recall Gandalf saying that a wearer did not die.

The "borrowing" is nowhere explicitly stated in the books, I don't think; that's just my theory of how the life-extension would work.
The Ring's power derives from Sauron, after all. By itself, it is only an object.

Originally Posted by Faramir Jones

I'm also a little confused. You say that 'when that influence is removed, nature takes its course'. Are you referring to the person ceasing to wear the Ring, or the Ring's destruction? While Bilbo, as you say, 17 years after giving up the Ring, was still old but able, he rapidly aged once the Ring was destroyed.

The Ring's destruction. As you note, Gollum too continued to live as long as the Ring survived. He told Sam on Mount Doom:

'And when Precious goes we'll die, yes, die into the dust'.

ROTK Mount Doom

I think Gollum knew that the Ring's loss would result in his physical death.