I figured out what happened, and I'm sorry for troubling the forum with what turned out to be a non-problem.
I have an MS Office keyboard with one of the buttons configured to launch Hotmail in my default browser. What happened is that Zone Alarm popped up with the notice that Avast Web Server was trying to contact the internet before that browser actually got launched. In other words, I was getting the notice before I could see my browser, and wondering why Avast was already asking to contact a dotted quad--just a matter of timing. I was thinking Avast was remembering my last visited website. Obviously, it was not.
Now that I've got that sorted out, I simply told Zone Alarm to remember the permission.
Thank you all.