I recently got a new computer (i7-860 and HD5770 video card) but my desktop is not full screen (about an inch of black all around). Im connected by HDMI, and the monitor is a HP2210m (max res 1920x1080, 70 Hz). The setup is currently at 1080p, 60 Hz.
Open Catalyst Control Center, and enable GPU overscan. I am sorry that I don't remember the exact process since I switched to nVidia, but if you search for how to enable gpu scaling ati 5770 you will find it. You will need to be able to slide a bar all the way to the left to fill up the screen, but I am sorry I don't remember the exact process, but I did have to do it a few times myself.