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) x 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.