I Know... That is What I Asked
Ben S.

"The Nazgul did not hold their rings, their rings held them :) Sauron had them, they did not wear them prancing around Middle-Earth, though they did of course bear them in their lives before they became wraiths. They were enslaved to Sauron through them. This is documented in Unfinished Tales and several other places I believe."

I know that, my question was what SAURON did with the ring when the Witch King bought the farm, so to speak.  I think that the Mouth of Sauron was the most likely candidate for it.