Plugged new mouse in again. Rebooted many zillion times. Fiddled much. Reinstalled new mouse's software many zillion times. Rebooted some more. Accomplished nothing, except that the computer now claimed to recognize the mouse's presence and calmly informed me that the mouse was functioning just perfectly, even though I couldn't move the cursor a single millimeter. Wondered if new mouse was on strike, holding out and refusing to work until I fed it cheese. Searched for mouse's mouth or any vaguely mouthlike orifice into which I could shove some cheese. Found nothing.
Plugged old mouse back in. Rebooted some more times. Voila! Old mouse now works again. New mouse does not.
At least next time the old mouse breaks again I'll have two of them to fiddle around with.