The end of Arwen Undómiel

This part of the story is present in an appendix of LOTR (meanings of some of the lesser known terms and persons are given below):

The Third Age ended thus in victory and hope; and yet grievous among the sorrows of that Age was the parting of Elrond and Arwen, for they were sundered by the Sea and by a doom beyond the end of the world. When the Great Ring was unmade and the Three were shorn of their power, then Elrond grew weary at last and forsook Middle-earth, never to return. But Arwen became as a mortal woman, and yet it was not her lot to die until all that she had gained was lost.
As Queen of Elves and Men she dwelt with Aragorn for six-score years in great glory and bliss; yet at last he felt the approach of old age and knew that the span of his life-days was drawing to an end, long though it had been. Then Aragorn said to Arwen:
"At last, Lady Evenstar, fairest in this world, and most beloved, my world is fading. Lo! we have gathered, and we have spent, and now the time of payment draws near."
Arwen knew what he intended, and long had forseen it, nonetheless she was overborne by her grief. "Would you then, lord, before your time leave your people that live by your word?" she said.
"Not before my time," he answered. "For if I will not go now, then I must soon go perforce. And Eldarion our son is a man full-ripe for kingship."
Then going to the House of the Kings in the Silent Street, Aragorn laid him down on the long bed that had been prepared for him. There he said farewell to Eldarion*, and gave into his hands the winged crown of Gondor and the sceptre of Arnor; and then all left him save Arwen, and she stood alone by his bed. And for all her wisdom and lineage she could not forbear to plead with him to stay yet for a while. She was not yet weary of her days, and thus she tasted the bitterness of the mortality that she had taken upon her.
"Lady Undómiel*," said Aragorn, "the hour is indeed hard, yet it was made even in that day when we met under the white birches in the garden of Elrond where none now walk. And on the hill of Cerin Amroth* when we forsook both the Shadow and the Twilight this doom we accepted. Take counsel with yourself, beloved, and ask whether you would indeed have me wait until I wither and fall from my high seat unmanned and witless. Nay, lady, I am the last of the Númenoreans and the latest King of the Elder Days; and to me has been given not only a span thrice that of Men of Middle-earth, but also the grace to go at my will, and give back the gift. Now, therefore, I will sleep.
"I speak no comfort to you, for there is no comfort for such pain within the circles of the world. The uttermost choice is before you, to repent and go to the Havens and bear away into the West the memory of our days together that shall there be evergreen but never more than memory; or else to abide the Doom of Men."
"Nay, dear lord," she said, "that choice is long over. There is now no ship that would bear me hence, and I must indeed abide the Doom of Men, whether I will or I nill: the loss and the silence. But I say to you, King of the Númenoreans, not till now have I understood the tale of your people and their fall. As wicked fools I scorned them, but I pity them at last. For if this is indeed, as the Eldar* say, the gift of the One to Men, it is bitter to receive."
"So it seems," he said. "But let us not be overthrown in the final test, who of old renounced the Shadow and the Ring. In sorrow we must go, but not in despair. Behold! We are not bound for ever to the circles of the world, and beyond them is more than memory, Farewell!"
"Estel*, Estel!" she cried, and with that even as he took her hand and kissed it, he fell into sleep. Then a great beauty was revealed in him, so that all who after came there looked on him in wonder; for they saw that the grace of his youth, and the valour of his manhood, and the wisdom and majesty of his age were blended together. And long there he lay, an image of the splendour of the Kings of Men in glory undimmed before the breaking of the world.
But Arwen went forth from the House, and the light in her eyes was quenched, and it seemed to her people that she had become cold and grey as nightfall in winter that comes without a star. Then she said farewell to Eldarion, and to her daughters, and to all whom she had loved; and she went out from the city of Minas Tirith and passed away to the land of Lórien, and dwelt there alone under the fading trees until the winter came. Galadriel had passed away and Celeborn also was gone, and the land was silent.
There at last when the mallorn-leaves were falling, but spring had not yet come, she laid herself to rest upon Cerin Amroth; and there is her green grave, until the world is changed, and all the days of her life are utterly forgotten by men that come after, and elanor and niphredil bloom no more east of the Sea.

Eldarion=son of Aragorn and Arwen
Undomiel=elvish for evening star, epithet of Arwen
Cerin Amroth=a beautiful hill (or mound) in Lothlorien
Eldar=elves
Estel=childhood name given to Aragorn by Elrond to hide his identity, means "hope"