FCC talking about relaxing cell phone rules

The news lately has mentioned that the FCC is meeting to discuss relaxing
the ban on using cell phones in airplanes. Most of the news coverage focuses
on the implications for airline passengers. They said that the FAA would
also
have to change rules before airline passengers could use cell phones.
The FCC is mainly concerned with interference with ground based use.
The FAA is mostly concerned about interference with aircraft electronics.
Does the FAA prohibit cell phone use in private aircraft, or is that just an
FCC rule? If it becomes legal to use a cell phone from the air, I may have
to look into one of those headset adapters for cell phones.