The Americans shall make the Saudis an offer they can't refuse:
There shall be a massive strengthening of the power and presence of the American Praetorian Guard in Saudi Arabia, and it will guarantee the stability of the throne. Also, the Americans will use their expertise to operate the oil fields at no extra charge.
In return, the Americans will be given permanent military bases in the kingdom, and the Saudi Royal Family will convert to Christianity.
Are there any other problems that need solving?
I really don't think they would have a great deal of difficulty refusing that offer