Thanks, will and PEU, for the suggestions. The set-up I was using was a four-jaw chuck mounted on and indexing head. In fact the workpiece was extended a fair bit from the chuck jaws, not to mention that the chuck was extended from the indexing head, so I think that will hit the nail on the head.
The tool was wicked sharp, so that wasn't part of the problem, nor was it sticking out from its collet in the headstock. I did have to rotate the workpiece by hand, and perhaps I just need to go slower, as well as figure out how to get it into the chuck further.
While I know, theoretically, how to grind a face grooving tool, I really didn't want to make one for such a tight, small groove, so I bought one instead. I may just give making one myself a shot, however.
Kenster: Yeah, buying a new tool is pretty much a commonplace occurence around here.