dr_peter
I concur with everything rudestuff & helilad have mentioned.
I did what you're planning and I would advise strongly against it. By all means do your FAA training there in short order time and with plenty of visa time left upon completion you'll have a very high chance of finding instructor / pipeline work to really boost your hours.
If you stick with with your combined JAA / FAA training plan you'll pay around $35 more for every hour you fly. Additionally you'll pay through the nose for ground school which many other folks have successfully self taught. All this will take you several months longer leaving your visa time thoroughly depleted. With little visa time left your chance of finding a flying job in the US is very slim indeed.
All that said - if you are indisciplined in self study then the JAA ground school may be for you. It was absolutely excellent - taught by first class instructors with tens of thousands of hours of real world experience.
Best of Luck
R1tamer