An X-ray detector has been flown on-board a NASA F-106 jet into thunderstorms. We have concluded that significant quantities of 3 to >12 keV X-rays are produced within thunderstorm clouds which support lightning distance activity. It is inferred that the X-rays are the bremsstrahlung radiation emitted from energetic electrons which have been accelerated through run-away processes. It is suggested and supported that the structures responsible for the acceleration are strong double-layers which are supported by the very high current densities attributed to lightning discharges.