No, no, it's not the whole computer that reboots, just Ubuntu, if that makes sense. It doesn't go to the grub screen or anything like that.
I took a look at that messages file (I had never seen that file before, it's pretty interesting). Nothing is listed at the exact time it restarts. For this particular time, I made sure that I went into the screensaver preview and left as quickly as I could, prompting the restart. So, Ubuntu started to restart at exactly 13:34:40. The log, though, doesn't list anything at that time, just when it began to restart--
Oct 23 13:34:26 andrew-laptop gconfd (root-7202): Resolved address "xml:readonly:/var/lib/gconf/defaults" to a read-only configuration source at position 4
Oct 23 13:34:46 andrew-laptop kernel: [ 552.248057] [fglrx] PCIe has already been initialized. Reinitializing ...
Oct 23 13:34:56 andrew-laptop gconfd (root-7202): GConf server is not in use, shutting down.
Oct 23 13:34:56 andrew-laptop gconfd (root-7202): Exiting