The problem was gone. Until just a couple of weeks ago.
I did a search and found this article:
Recommending this fix:
But the fix is for Win7 Beta and an earlier version WMP.
Anyone know of a fix for WMP 12 and/or for a fully updated Win7?
Obviously, the best fix is to no longer use WMP for MP3s, which I've now done.
(I hate WinAmp and most of the more-popular alternatives and only just recently discovered Quintessential Media Plater.)