That sounds like the old ATI CCC underscan/overscan bug.
Goto CCC (Catalyst Control Centre / AMD Vision) -> My Digital Flat-Panels -> Scaling Options
Make sure the Underscan/Overscan slider is set to 0%.
Note.
I believe this is a per-resolution/refresh rate setting, so you'll need to make sure Skyrim is running at the same resolution & refresh rate as your desktop for the setting to 'stick'.
If you haven't encountered this issue in other games, then I'd guess Skyrim is (for whatever reason) running in a different resolution/refresh rate. (perhaps 1920x1080@59Hz ?)
Goto CCC (Catalyst Control Centre / AMD Vision) -> My Digital Flat-Panels -> Scaling Options
Make sure the Underscan/Overscan slider is set to 0%.
Note.
I believe this is a per-resolution/refresh rate setting, so you'll need to make sure Skyrim is running at the same resolution & refresh rate as your desktop for the setting to 'stick'.
If you haven't encountered this issue in other games, then I'd guess Skyrim is (for whatever reason) running in a different resolution/refresh rate. (perhaps 1920x1080@59Hz ?)
Yeap, that's it! everything is perfect now... even the desktop! Thank you very very much! and all of you from the forum for your time and effort! At last I will enjoy the game at least until I get an arrow in the knee!

