I want to replace an existing 2-prong receptacle in a bathroom with a GFCI
receptacle. There is no ground wire going to the existing receptacle and no
grounded metal cable or grounded metal outlet box -- just two separate
wires, one white and one black. There is also no practical way to run a new
ground wire back to the service.

In this situation, is it okay to go ahead and install the GFCI receptacle
and put the sticker on it that says "No Equipment Ground"?

UP.SO. I "could" run a ground wire from the GFCI receptacle to a cold water
pipe and make sure that there is a jumper across the hot water tank in and
out pipes and across the water meter. But, I think I read somewhere about
that being a bad idea (using the cold water pipe for a ground). That is a
bad idea, right?