Having done exactly that on different computers (Dell XP OS on my custom built PC, a Compaq XP OS for Dell PC, and a Acer Vista Basic on, erm, well, Acer) it should work. Just so long as it is the actual OS itself, not a restore disk. Just make sure you have a key for it (any key should work, but sometimes Microsoft is on the ball and prevents this).
Now, there is always a chance Dell has wisened up and made it for them only, but its always been my experience that this isn't ever the case.