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.