I got it!
I found this page: http://forlong.blogage.de/article/pages/Compiz-Check
I downloaded it and ran it, and it told me exactly why my card wasn't working: compiz was blacklisting it. The program then removed the blacklist, and it ran beautifully.
Thanks, everyone.
Code:
$ ./compiz-check
Gathering information about your system...
Distribution: Ubuntu 8.04
Desktop environment: GNOME
Graphics chip: ATI Technologies Inc Radeon IGP 330M/340M/350M
Driver in use: radeon
Rendering method: AIGLX
Checking if it's possible to run Compiz on your system...
Checking for texture_from_pixmap... [ OK ]
Checking for non power of two support... [ OK ]
Checking for composite extension... [ OK ]
Checking for FBConfig... [ OK ]
Checking for hardware/setup problems... [FAIL]
There has been (at least) one error detected with your setup:
Error: Laptop using radeon driver.
Would you like to know more? (Y/n) Y
It has been detected, that you are running a laptop with an ATI chip.
The radeon driver supports Compiz out-of-the-box but because of a nasty bug
in the driver that causes X to freeze on some cards, this particular
combination had to be blacklisted in Ubuntu "Hardy Heron".
In case you already used Compiz successfully on Ubuntu 7.10 (Gutsy), it is
safe to skip the blacklist.
Do you want to skip blacklist checks by Compiz? (y/N) y
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