Here are the Author's Notes for "Testaments of the Past", Chapter 6, "To Cast Away a Treasure at Need..."
CHAPTER SIX AUTHORS' NOTES
CHAPTER 6
Here is a picture of the chain in the Gondorian box that Sam, Merry, and Pippin found in the strongbox:
AUTHOR'S NOTES:
It was important to us in writing this story that Sam, Merry and Pippin be equally surprised that the chain had survived Mount Doom. Their speculation is close to the mark, although of course, it doesn't explain how Frodo came to have the chain in his possession. However, we felt that you, the readers, deserved the full explanation, even if Sam, Merry and Pippin do not learn it.
The chain was indeed, caught in Frodo's hair, and the matted blood where it had cut into his neck. Of course, when he was brought back to the healers, it was removed, and put aside with the other things they yet had. Gandalf and Aragorn questioned what was to be done with it. Aragorn thought it should be disposed of without Frodo ever knowing, but Gandalf thought Frodo deserved the chance to choose what he wished done. He hoped that Frodo would also choose to dispose of it, and thought that giving Frodo the chance to *make* that choice would help him. However, Frodo decided to keep it, and Aragorn provided him with the small box to put it in.
It's possible that at some future point in time these three will think to ask Aragorn about the chain.