Does Frodo Baggins die in The Lord of the Rings?

In the appendices of The Lord of the Rings, Frodo, because he was a Ring Bearer sailed to the Undying Lands to live out the rest of his days in the purity of Valinor. The book does not say when he died. Sam was also accorded the same honor.

It is generally assumed that the hobbits that were Ring Bearers (Bilbo, Frodo, and Samwise) never died, but were granted eternal life as the other residents of the Undying Lands had. Of course for this to be true they must have been fully healed of all injuries both due to bearing the ring and ordinary life, both physical and mental/spiritual. However as stated above, the books don't say so nothing can be said for sure.