Open the Ubuntu Software Center and search for lib32. If that turns up any results, you are on a 64-bit install (the results are compatibility libraries for running 32-bit applications on a 64-bit install).
Not exactly a better answer, but at least it doesn't require a terminal... ;-)
Edit
I found an even easier one: open Help -> About Mozilla Firefox and you will see it right there... ;-)
At the bottom it displays the "user agent string", e.g. on my 64-bit system:
Mozilla/5.0 (X11; U; Linux x86_64; nl; rv:1.9.2.12) Gecko/20101027 Ubuntu/10.10 (maverick) Firefox/3.6.12
or on my 32-bit system:
Mozilla/5.0 (X11; U; Linux i686; nl; rv:1.9.2.12) Gecko/20101027 Ubuntu/10.10 (maverick) Firefox/3.6.12
x86_64 is 64-bit, i686 is 32-bit
(But this is still not the answer that should be there ;) )