The STAR Collaboration at the Relativistic Heavy Ion Collider reports measurements of azimuthal correlations of high transverse momentum (pT) charged hadrons in Au+Au collisions at higher pT than reported previously. As (pT) is increased, a narrow, back-to-back peak emerges above the decreasing background, providing a clear dijet signal for all collision centralities studied. Using these correlations, we perform a systematic study of dijet production and suppression in nuclear collisions, providing new constraints on the mechanisms underlying partonic energy loss in dense matter.