Thanks, guys, but I see now that it wasn't adware or malware...
I have a commercial FTP site access program called WebDrive. I had a drive connection established to my web site files, and the FTP program normally sends "keep alive" messages occasionally, for obvious reasons.
Trouble is, my router spontaneously rebooted (something it doesn't normally do and that I need to get to the bottom of). Apparently one of the libraries deep inside WebDrive sent the pop-up pictured above. When I OK'd the pop-up another message came up from WebDrive, which showed that WebDrive was blocked on the first pop-up and confirmed the diagnosis.
The Avast! scan came up clean, by the way, and I've confirmed I have no adware on this system.
-Noel