Unless you already have a JAA PPL and can get it validated, you will need an FAA Medical (with student pilot certificate) so that you can go solo in US airspace.
FAA Medicals are expensive in Europe - they're only $80 over here!
You are also required to have a current JAA medical if taking the JAA PPL skills test so bare that in mind if you travel to the states to do a JAA PPL.