On the odd chance that this is not a troll (which I suspect it is)...
You don't need a three-week period to obtain a standalone PPL. A few days should be sufficient. All training leading to your Belgian PPL is recognised as valid so what you will only need to do is:
- Do the single FAA PPL written exam (note that this might require some preparation, particularly some intimate time with the FAR-AIM)
- Get some refresher training, particularly on the few maneuvers that are in the FAA syllabus but not the JAA syllabus. E.g. turns around a point
- Do the five hours plus the solo stop-and-goes for the night qualification
- Do the FAA PPL skills test
All this should be doable in just a few days.
Do note that in all likelihood you'll need an M-1 visa and TSA clearance.