Mortals in Aman
Based on Barrow-Downs Forum material compiled (and mostly written) by Legolas
Any errors of spelling or grammar have been corrected silently.

"At the end of Lord of the Rings, I kept on wondering. What happened to Sam?" (-Maikacairien)

"Who actually thinks that Frodo and the other ring bearers made it to Valinor considering in the Silmarillion it says 'In Aman the Blessed Realm was shut against the Noldor and of the the many messengers that in after days sailed into the west none came ever to Valinor'?" (-Alatar_69)

"The situation changed at the end of the War of Wrath, when most of the surviving Noldor were accepted back into the Undying Lands, and again when the world was "bent" at the time of the Fall of Numenor. The rules weren't fixed; the Valar could admit or exclude anyone they pleased." (-Selmo)

Frodo, Bilbo, Sam, and Gimli journeyed to the Undying Lands for good, along with Elrond, Gandalf, and Legolas. These four mortals (Frodo, Sam, Bilbo, Gimli) lived there for a while and died, as indicated in the last lines of the second quote and in the following quotes presented here. Exceptions had obviously been granted...it was a long time after the said ban on emigration was announced.*

Tolkien addressed the matter of the mortals reaching Aman in a few of his letters, each explaining that the ringbearers (and Gimli) found peace there, but eventually passed away, just as all mortals had to:

"But in this story it is supposed that there may be certain rare exceptions or accommodations (legitimately supposed? there always seem to be exceptions); and so certain 'mortals', who have played some great part in Elvish affairs, may pass with the Elves to Elvenhome. Thus Frodo (by the express gift of Arwen) and Bilbo, and eventually Sam (as adumbrated by Frodo); and as a unique exception Gimli the Dwarf, as friend of Legolas and 'servant' of Galadriel. I have said nothing about it in this book, but the mythical idea underlying is that for mortals, since their 'kind' cannot be changed for ever, this is strictly only a temporary reward: a healing and redress of suffering. They cannot abide for ever, and though they cannot return to mortal earth, they can and will 'die' of free will, and leave the world (Letter 154)."

"'Alas! there are some wounds that cannot be wholly cured', said Gandalf (III 268) not in Middle-earth. Frodo was sent or allowed to pass over Sea to heal him if that could be done, before he died. He would have eventually to 'pass away': no mortal could, or can, abide for ever on earth, or within Time. So he went both to a purgatory and to a reward, for a while: a period of reflection and peace and a gaining of a truer understanding of his position in littleness and in greatness, spent still in Time amid the natural beauty of 'Arda Unmarred', the Earth unspoiled by evil (Letter 246)."

"As for Frodo or other mortals, they could only dwell in Aman for a limited time - whether brief or long. The Valar had neither the power nor the right to confer 'immortality' upon them. Their sojourn was a 'purgatory', but one of peace and healing and they would eventually pass away (die at their own desire and of free will) to destinations of which the Elves knew nothing (Letter 325)."

Primary sources:
The Letters of J.R.R. Tolkien (154, 246, 325)
Lord of the Rings

Forum resources:
On the Ring-Bearers' Fate I*
On the Ring-Bearers' Fate II