Very odd. Right-click on the Firefox icon on your desktop and check the properties for any clues, such as General info, the file details and Shortcut target, then post back.

You could also download a fresh new copy of the Firefox installer file "Firefox Setup 10.0.2.exe" from http://www.mozilla.org/firefox/new and save it to your desktop. Next, uninstall Firefox from the Windows Control Panel Add or Remove Programs list, to see if that gets rid of the Firefox icon. (When you uninstall Firefox, it will not remove your bookmarks or other user data and settings as long as you don't check that option.) Ref: Uninstall Firefox from your computer

You can then reinstall Firefox by running the setup file you downloaded. Ref: How to install Firefox on Windows

Thanks, Alice. I have 10.0.2 already. This is odd indeed. Every day since upgrading, I get that icon back on my desktop, and I delete it. I take bets as to when it decides it needs to come back! I thought maybe there's a setting that was added that some Firefox marketing whiz came up with. I'm in the dark on how to fix, but no big deal. I can live with it I guess. Maybe someone else will post a response. Cheers

Your User Agent actually shows Firefox 10.0 (not 10.0.2):

Mozilla/5.0 (Windows NT 5.1; rv:10.0) Gecko/20100101 Firefox/10.0

...so you should update to 10.0.2 if you haven't already - see Find what version of Firefox you are using.

In any case I would have suggested an uninstall -> reinstall, in case uninstalling Firefox would help get rid of that Firefox desktop icon that won't stay deleted.

as I said " I have 10.0.2 already. "

And I am concerned about uninstalling Firefox as I don't know what I'd lose (Favorites, stuff in memory, passwords, etc). Not a leap I want to forgo without knowing the downside first.

Uninstalling Firefox won't touch your favorites (bookmarks) passwords or any other Firefox personal data or customizations as long as you don't checkmark the "Remove" box. See the attached screenshot.