LotR, FotR, Book 1, Chapter 1. Bilbo says,
It's mine, I tell you. My own. My precious. Yes, my precious."
Isildur was beginning to have the same problem. LotR, FotR, Book 2, Chapter 2.
But for my part I will risk no hurt to this thing: of all the works of Saruon the only fair. It is precious to me, though I buy it with great pain.
Frodo was also coming under the Ring's spell. (as already quoted by Perwing).
Saruman was even coming under the spell of the Ring (since he had studied it and become obssessed with it), so he also called it Precious.
LotR, FotR, Book 2, Chapter 2
... I believe that you know this precious thing now lies.
Tom is about the only anomoly, but that's just Tom's nature anyway.
I don't think there is much difference between My Precious and The Precious Ring. I think it still means that the Ring is influencing the person who thinks/says it is Precious.
Just some more of my thoughts. I love reading what everyone thinks of this. I just can't believe I never noticed it all the other times I've read LotR.
[Edited on 5/5/2003 by McDLT]