I did my PPL in 5 weeks in KFXE, which turned out flying nearly every day. I would guess that 3 weeks is too short, since you have to calculate some bad days (due wx) and some pratice time (a friend of mine tried to do it in 6 weeks and was unable, since he got sick). Beware of a school that promises you it may be done in that time without any problem (the only one having completed it within this time is probably the son or daughter of the instructor - since they obviously already spend some time in the right seat before
).
I did mine at Sheffield School of Aeronautics (since I already knew the staff). I was pretty happy with the setup, however, there is a whole bunch of schools out there.
Cheers
fcit