No, according to his obituary, Frodo did not die.
He died in the West.
In Tolkein’s world, elves were naturally immortal did not die naturally. If they were killed, they woild be reborn with all of their memories intact.
Man and halflings were not immortal and could not become immortal. They would die after living their pre-determined lifespan, even if they lived in a place where things didn’t naturally die.