Who gave Frodo a mithril coat?

Bilbo Baggins gave Frodo a mithril coat before Frodo began his quest. Bilbo had originally received the small coat of mail from Thorin Oakenshield after the events of The Hobbit.

The shirt had been recovered from the hoard of Smaug and was made from mithril, an exceptionally light, strong, and rare metal. Bilbo wore it during the Battle of the Five Armies and later reclaimed it after placing it in the Mathom-house museum in Michel Delving. He eventually brought it to Rivendell and gave it to Frodo.

The coat proves vital during the journey. Frodo wears it beneath his clothing, and it saves his life when he is stabbed by an Orc in Moria. Its protective value is emphasized when Gandalf says that the shirt is worth more than the entire Shire and everything in it.

The common mix-up is Thorin: Thorin gave the mithril coat to Bilbo, but Bilbo was the person who later gave it to Frodo. Elrond, Aragorn, and Gandalf did not give Frodo this particular gift.