This is the proper command to install the accelerated ATI drivers on Linux Mint Debian Edition. The fglrx-driver package allowed me to use higher resolutions after enabling the driver with this command:
sudo aticontrol --initial but the driver did not allow acceleration.
After using this command to install all of the required ATI packages, I got the proper accelerated desktop working.
sudo apt-get install fglrx-driver fglrx-source fglrx-control fglrx-modules-dkms
This is what you should see when you have this working.
homer@deusexmachina ~ $ glxgears libGL error: failed to load driver: swrast libGL error: Try again with LIBGL_DEBUG=verbose for more details. Running synchronized to the vertical refresh. The framerate should be approximately the same as the monitor refresh rate. 25434 frames in 5.0 seconds = 5086.092 FPS XIO: fatal IO error 11 (Resource temporarily unavailable) on X server ":0" after 75810 requests (65177 known processed) with 0 events remaining.