Who had hung the ring about Frodo's neck on a new chain?

Question

'Have you got it here?' Bilbo asked in a whisper. 'I can't help feeling curious, you know, after all I've heard. I should very much like just to peep at it again.'

(...)

When he had dressed, Frodo found that while he slept the Ring had been hung about his neck on a new chain, light but strong.

Who had done that?

As the quote shows, it wasn't Bilbo, since he didn't know Frodo had it on him. And as the scene continues we see Bilbo being tempted by the ring, so we can be sure it wasn't Bilbo handling the ring before.

Answer 1

There's no mention.

The last we hear of the Ring before Frodo wakes in Rivendell is just after he is attacked on Weathertop:

When Frodo came to himself he was still clutching the Ring desperately.

Book I, Chapter 12, "The Flight to the Ford"

At this point, it's still on its chain; during the attack Frodo wears it this way.

Resistance became unbearable, and at last he slowly drew out the chain, and slipped the Ring on the forefinger of his left hand.

Book I, Chapter 11, "A Knife in the Dark"

We see no more of the Ring until the passage you mention, and there is no mention of who provided the "clean garments of green cloth" in which Frodo dressed when he got up. Presumably it was at the orders of Elrond, who seems to have been Frodo's primary caregiver; it seems unlikely he would have told assistants about the Ring. (Even Frodo guessed it was not a matter for casual talk.)

There's no further mention of the Ring in The Shadow of the Past, that part of the History of Middle-earth which discusses early drafts of The Fellowship of the Ring. In fact this particular scene with Bilbo and Frodo doesn't appear in that volume.

So we are left guessing. Perhaps most likely is that it was an assistant of Elrond, an inhabitant of Rivendell who was unaware that this was The Ring.

Answer 2

No idea at all, but we could make accurate assumptions.

1. Elrond, Lord of Rivendell

   It's likely. Remember that Elrond is the one to care for Frodo after the events of "Flight to the Ford". He's the one that removes the splinter blade from Frodo's shoulder.

   'Good!' said Gandalf. 'It is mending fast. You will soon be sound again. Elrond has cured you: he has tended you for days, ever since you were brought in.'

2. Gandalf

   Gandalf seems to have been sitting at Frodo's side most of the time.

   'Where am I, and what is the time?' he said aloud to the ceiling. 'In the House of Elrond, and it is ten o'clock in the morning.' said a voice. 'It is the morning of October the twenty-fourth, if you want to know.'

   'Gandalf!' cried Frodo, sitting up. There was the old wizard, sitting in a chair by the open window.

   'Yes,' he said, 'I am here. And you are lucky to be here, too, after all the absurd things you have done since you left home.'

Other possible guesses:

3. Elvish caretakers of Frodo