This article is from the Tolkien FAQ, by William D.B. Loos loos@hudce.harvard.edu with numerous contributions by others.
They remained mortal. Tolkien's conception was that a creature's natural
lifespan was intrinsic to its spiritual and biological nature, and that this
could not be altered save by a direct intervention of the Creator. There
were three occasions when this did happen (Luthien, Tuor, Arwen), but it did
not in the cases of Frodo & Co. Tolkien stated explicitly in more than one
letter that Frodo's journey over the Sea was only a *temporary* healing, and
that when the time came he and the others would die of their own free will.
 
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