Make sure you get a STAND ALONE FAA certificate and NOT one based on a foreign licence - they are a waste of time.
You need to set up an account with the TSA and do your fingerprints,
https://www.flightschoolcandidates.gov which also has some useful info.
You also need to do an ATP written exam, I recommend the ASA test prep book, you have a choice out of part 121 or part 135, either will work without ever limiting your future options. 135 is probably more relevant to what you're ever likely to do with the Gulfstream.
Before you take the check, you'll need an FAA medical in addition to your JAA one. You also need to fill in any experience blanks listed in part 61 but in practice I doubt there will be any with your current licence.
The cheapest way of doing the checkride (Like an LST) is to combine it with your next recurrent. It's worth spelling out to them that you additionally want an ATP initial and a Type initial check. There is an oral test too, most examiners worth their salt will make the oral about the type in question in these circumstances but ANY theory issue is fair game and given that your new to the FAA system I'd say it would be prudent to give the commercial test prep a look too, and get to know your way around the FAR-AIM. Nothing arduous here just sensible wide-reading. Anything you got wrong on the written is printed on the test report and you WILL be asked about this.
Assuming that all goes well you'll be issued there and then with a temporary airmen certificate and you're good to operate immediately.
If you need help with where to go to get all this done (most can be sorted in europe) or combining with a recurrent PM me with your location and email and I'll hook you up with some contacts.
It sounds a lot but there is nothing really costly or difficult here. I have both JAA and FAA, welcome to the common sense system!
Regards
Jeff