I have two Ubuntu machines, a desktop that has a
32-bit CPU and a laptop that has a dual-core 64-bit CPU. The command
"uname -a" gives "Linux jim-desktop 2.6.27-7-generic #1 SMP Thu Oct 30
04:18:38 UTC 2008 i686 GNU/Linux" for the desktop and the same info for
the laptop (except for jim-laptop).

Since they both say the same thing, must be that they are both running a
32-bit OS.