The Mysterious 10th Bearer.

[Keeper of Dol Guldur]

I don't know if it's been covered already, but I honestly think it was Elrond who changed the chain.

While he was healing Frodo, personally, he would have been undressing Frodo, and I doubt he would leave such an heirloom and menace in the pocket of Frodo's overcoat where it could actually if it felt like it, fall out haphazardly, roll down the floor and by some chance of luck, off a cliff into the Bruinen, to be washed down to the Ringwraiths as they collected themselves on the banks miles down, without clothes.

My point being, it couldn't be trusted to just be left in the pocket, and I doubt he would burden / trust (use these terms lightly, we all know the formalities and tricks that the ring requires people to take into account) any other elf with moving it, or touching it, or handling it in any way. He knew the best place for it was around Frodo's neck, where it couldn't make an exit, tricky thing.

While it's possible as he switched it, he had a Galadriel like test of whether he should take it or not, I doubt it.

For starters, Elrond wanted the thing destroyed, he had for thousands of years, ever since Isildur got a hold of it.

Second, Gandalf would likely have been there with him. There's strength in numbers, and reasoning too. Especially since they both had Elven rings. Anyway, it would have been impossible.

No, I imagine he handled the thing, just like Gandalf had done when he threw it into Bag End's fireplace, many weeks before. He probably quickly and gingerly grabbed it, looped the chain in, and had as short a contact with the thing as possible.

So I'm assuming it was Elrond.

It's definitely true that handling it and bearing it are completely different.