Using a NaI(Tl) scintillation detector designed to operate in electrically noisy environments, we observed intense bursts of energetic radiation (>> 10 kiloelectron volts) during the dart leader phase of rocket-triggered lightning, just before and possibly at the very start of 31 out of the 37 return strokes measured. The bursts had typical durations of less than 100 microseconds and deposited many tens of megaelectron volts into the detector. These results provide strong evidence that the production of runaway electrons is an important process during lightning.