How can I permanently get rid of a desktop icon?

The offending shortcut is probably in the Desktop folder at the All User profile rather than in your profile. This being the case, even though you delete the shortcut from your desktop view, you're not really deleting it, you're just removing it from the current view, and whenever you restart the machine, the All User profile pops it right back up on the desktop. To really delete the shortcut, right-click on the Start on, select Explore All Users, go to Doents and Settings, All Users, open the Desktop folder, find and delete the shortcut, then close the Doents and Settings folder. Next time you restart, the shortcut will not appear.