The PSR B1259-63 system (Johnston et al. 1992, 1994) was observed near periastron by the Compton Gamma-Ray Observatory in 1994 January. This system contains a rapidly rotating pulsar and a Be star in a highly eccentric binary orbit. We report the discovery by the OSSE instrument of unpulsed emission with a hard power-law spectrum between 50 and 200 keV from the direction of this system. Neither diffuse Galactic background emission nor nearby X-ray binaries contribute significantly to the detected flux. Our results are particularly important for the theory of interaction of pulsars with gaseous environments. We interpret the hard X-ray emission as synchrotron radiation from relativistic particles of the PSR B1259-63 wind being shocked and accelerated within the binary. Our results indicate, for the first time in a binary pulsar, that shock acceleration can increase the original energy of pulsar wind particles by a factor greater than or similar to 10, despite the high synchrotron and inverse Compton cooling rates near periastron. The derived shock properties (efficiency, radiation spectrum, timescale) are relevant for a broad class of high-energy astrophysical sources characterized by shocked relativistic plasmas subject to strong radiative cooling.