In Tolkien's universe, it's not Elves, but Men (including Hobbits) who are truly immortal.
Elves will exist only as long as the world exists. The spirits of Elves who are killed go to the Halls of Mandos in Amman which, although separated from Middle Earth, is still part of the physical world. When the physical world comes to an end, so do the Elves.
Men, on the other hand, are physically mortal but spiritually immortal. After death their spirits can leave the circles of the world and will continue to exist after the end of the world.
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