Although I haven't done what you're asking, I did once get a based-on (§61.75) FAA Private Pilot certificate when I went there to hour-build. I went to Chandler Air Service for 3 weeks, and highly recommend them.
Then, a year or two later, when I got my FAA "standalone" Commercial (also at Chandler), it superseded the Private and then I became pure FAA and could forget about SEP ratings and all that bollocks having to keep my underlying JAA licence current. I was able to still rent G-reg's back in the UK for flying with PPL privileges, purely on my FAA certificate.
There is plenty of information on here and elsewhere on the web about how you go about getting a based-on FAA certificate, so I won't go into that.
You will need a visa, and TSA clearance if you intend to do any training towards a certificate or rating, so start lining all that up if that's the route you intend to go down. If you're only going to do a bit of flying, then you should be ok on an ESTA.
To obtain a standalone FAA Private, then it's just a case of meeting the experience requirements detailed in the regs (and your JAA hours/experience will count) which shouldn't take too long and will include night flying (if you don't have a Night Qualification) and learning how to fly a few "ground reference manoeuvres" not covered in your JAA syllabus.
When will you be going?