Gandalf does touch the One. He looks at it, then throws it manually into Frodo's fireplace.
There is whoever takes the Ring from Frodo's pocket, takes it off its old chain and puts it on a new one and around Frodo's neck in Rivendell. They could have hooked it with a stick, I suppose, but ....
My money would be on Elrond.
Reminds me of early themed passwords and spelling Khamul backwards, or something.
But the last Oxonmoot Quiz had a questiom like this too, IIRC.