[Man-of-the-Wold]

First, I agree with others there are only six persons who ever really possessed or "bore" the One Ring: Its maker, Sauron; the man to whom it was his bane, Isildur, and four hobbits--Smeagol, Bilbo, Frodo and Sam.

I consider the best theory is that one or two unnamed elves who attended to Frodo and assisted Elrond, simply removed the Ring from Frodo's clothing at Elrond's instructions, put aside, and then the chain was either summoned or fashioned and brought to the sick room, at which time Elrond, Gandalf or again an unnamed elf fastened it around Frodo's neck. Most any Elf in attendance would not have known what it was and would not have been affected in any profound way like a man might if briefly handling the Ring. Bilbo couldn't have done. If Sam had, it would have been mentioned at some point. Elrond certainly could have done it, but likely would've wanted not to touch so unclean a thing. Gandalf obviously could have done it and was not opposed to momentary handling of it.