If you're using the default trigger for autofocus (holding the shutter release button halfway down) and you let it go after focusing, then recompose, then push the shutter release button again, the camera will indeed refocus on whatever the active focus point is aimed at.
In order to focus/recompose, you need to somehow deactivate the autofocus system between the focus action and tripping the shutter after recomposing. The default way is to HOLD the shutter release button halfway down for the duration of the time between focusing and tripping the shutter.
Many of us have our cameras configured to use a button on the back of the camera (the * button in the case of my 20D) to activate auto focus. With that method, I can aim the camera so the center focus point is on whatever I want the camera to focus on, tap the * button, then do whatever I want until I trip the shutter. Without tapping the * button again, the focus will essentially be in manual focus mode and won't change by itself.
There is also a way to lock the focus, but I find that method (which I cannot remember) a pain in the you-know-where to mess with. Just using the * button whenever (and IF) I want to activate autofocus works great for me. I can, for example, take a whole series of shots with the same camera-to-subject distance and only set the focus once. I can also tweak the focus manually any time I want (with the appropriate lenses that don't have a gear train that you'd be stressing doing it).