And so it was that Bilbo and Frodo sailed to The Undying Lands as father and son. By the time they reached Tol Erresea, the years had fallen away from Bilbo, and he looked again like a 50 year young Hobbit. Elrond and Gandalf took the pair to the mainland to meet Manwe.
Though it was not within his power to bestow, Manwe had petitioned Eru Illuvatar, and He had granted the Hobbits a special dispensation for bearing the One Ring—and thus Sauron—to Doom. In all the Millennia, none other had been up to the task. Frodo being Half-Elven was allowed to choose an Immortal life. And Bilbo—Elf-friend—was granted status as 'honourary' Elf, and an immortal life. For Hobbits—being of Stout Heart—could bear it.
Bilbo and Frodo then traveled to the Gardens of Lorien; and there awaiting her Beloved, was Tauriel, as radiant and full of life as the day Bilbo first met her. And thus, the Baggins family was made whole, to enjoy the delights of Valinor for Eternity.
And having come to know his True Mother, and thus discovering the roots of his destiny, Frodo finally was whole. And he eventually came to marry Astarte, daughter of Galadriel and Celeborn.
"So that's why Galadriel always looked at me like that." Frodo mused on his wedding night.