I don't think that Bilbo placed the Ring on the chain because the text in the chapter entitled Many Meetings states:
Quote:
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'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.'
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If Bilbo had handled the Ring in placing it on a chain and hanging it around Frodo's neck...why would he have asked this?
The ring was probably placed around Frodo's neck because he wasn't wearing his clothes while in bed...therefore no pocket to keep the ring in! Maybe the old chain wasn't long enough to go over his head.
I think that either Gandalf or Elrond switched it.
By the way...when did Gandalf bear the ring...I simply cannot remember a time that he did!!!