We report the observation of two isolated clouds of positrons inside an active thunderstorm. These observations were made by the Airborne Detector for Energetic Lightning Emissions (ADELE), an array of six gamma-ray detectors, which flew on a Gulfstream V jet aircraft through the top of an active thunderstorm in August 2009. ADELE recorded two 511 keV gamma-ray count rate enhancements, 35 s apart, each lasting approximately 0.2 s. The enhancements, which were approximately a factor of 12 above background, were both accompanied by electrical activity as measured by a flat-plate antenna on the underside of the aircraft. The energy spectra were consistent with a source mostly composed of positron annihilation gamma rays, with a prominent 511 keV line clearly visible in the data. Model fits to the data suggest that the aircraft was briefly immersed in clouds of positrons, more than a kilometre across. It is not clear how the positron clouds were created within the thunderstorm, but it is possible they were caused by the presence of the aircraft in the electrified environment.