The general rule is that you can't fly as PIC of an aircraft registered in one ICAO State on the license of another, however there is a process for "rendering valid" a foreign license to allow it. In the US it's done by issuing an FAA license which is only valid if your underlying foreign license is valid.
The bit of LASORS you want is this
A3 BASIC AUTHORITY TO ACT AS A
FLIGHT CREW MEMBER
A person shall not act as a flight crew member of a civil
aeroplane/helicopter registered in a JAA Member State
unless that person holds a valid licence and rating
complying with the requirements of JAR–FCL and
appropriate to the duties being performed, or an
authorisation as set out in JAR–FCL 1.085/2.085 and/
or 1.230/2.230. The licence shall have been issued by:
• A JAA Member State; or
• Another ICAO Contracting State and rendered
valid in accordance with JAR–FCL 1.015/
2.015(b) or (c).
JAR-FCL can be found
here.
Alternatively you can get a stand-alone JAA License
The holder of a current and valid PPL(A)* issued
by an ICAO Contracting State (not being a JAA
Member State), who has flown a minimum of 100
hours as pilot of aeroplanes, is credited the
JAR-FCL PPL(A) flying training/experience
requirements, except the PPL(A) Skill Test.
(from LASORS)
So it looks like you just do the skill test and get your license.