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Starburst galaxies exhibit in their central regions a highly increased rate of supernovae, the remnants of which are thought to accelerate energetic cosmic rays up to energies of ~1015 electron volts. We report the detection of gamma rays—tracers of such cosmic rays—from the starburst galaxy NGC 253 using the High Energy Stereoscopic System (H.E.S.S.) array of imaging atmospheric Cherenkov telescopes. The gamma-ray flux above 220 billion electron volts is F = (5.5 ± 1.0stat ± 2.8sys) × 10−13 cm−2 s−1, implying a cosmic-ray density about three orders of magnitude larger than that in the center of the Milky Way. The fraction of cosmic-ray energy channeled into gamma rays in this starburst environment is five times as large as that in our Galaxy.