Book II, Chapter 1: "Many Meetings"
Frodo awakens to in Rivendell, Gandalf by his bed. The wizard explains that he could not reach Frodo because he was being held captive. He also tells Frodo that the shard of the Ringwraith's blade has been removed from his body; had it reached Frodo's heart, he would have become a wraith as well, enslaved to Sauron. Frodo asks if the flood at the Ford-summoned by Elrond, with "a few touches" such as the horses created by Gandalf-means the end of the Ringwraiths. Gandalf informs him that the wraiths have not been destroyed.
After a reunion with each other, Frodo and the other Hobbits join a feast in Elrond's hall. At the feast, Frodo visits with Glóin, one of the Dwarves who accompanied Bilbo on his quest to the Lonely Mountain so many years before. Frodo asks Glóin about the fate of Balin, another of Bilbo's companions. Glóin reveals that he does not know what has happened to Balin, which is one reason he has come to Rivendell, to seek the wisdom of the Elves in that matter.
After the feast, Frodo is reunited with Bilbo himself. Bilbo asks if he might see the Ring again. When Frodo shows it to him, Bilbo seems to him "a little wrinkled creature with a hungry face and bony groping hands." Tolkien's descriptive language is significant because it not only further illustrates the Ring's power to corrupt, but also prepares readers to meet Gollum (from whom Bilbo took the Ring in The Hobbit) in Book IV: Gollum matches this description exactly.
Bilbo tells Frodo that he has become friends with Aragorn. Bilbo recites a long poem he has composed about the Elves of the Elder Days.
Before retiring for the night, Frodo spies Aragorn and Arwen, Elrond's daughter, talking together. "[S]uddenly it seemed to Frodo that Arwen turned towards him, and the light of her eyes fell on him from afar and pierced his heart" (cf. Frodo's reaction to Goldberry in Book I, Chapter 7.)