What video card ati or nvidia? What your looking for is the scaling options, on AMD cards you will find this in the Catalyst Control Center under - My digital flat panels - properties. From there you can adjust the scaling to get rid of the black border.
Not sure where the option is in the Nvidia control panel but it does also have a section for adjusting the monitor scaling.
Also you might find that using a different video cable will help, ie if your using HDMI currently then try a DVI-D cable if you have the supported ports on your monitor and video card.
I had the same problem while ago when my screen was connected by HDMI, fixed by adjusting the scaling options, later I switched to display port cables and I didn't have to bother...