We report the results from X-ray and hard X-ray observations of the PSR B1259 - 63/SS 2883 system with the ASCA and CGRO satellites , performed between 1995 February and 1996 January when the pulsar was near apastron . The system was clearly detected in each of the two ASCA observations with luminosity in the 1–10 keV band of L _ { X } = ( 9 \pm 3 ) \times 10 ^ { 32 } ( d / 2 { ~ { } kpc } ) ^ { 2 } erg s ^ { -1 } , while CGRO/OSSE did not detect significant hard X-rays from the system . X-ray spectra obtained with ASCA are well-fit by a single power-law spectrum with a photon index of 1.6 \pm 0.3 . No pulsations were detected in either the ASCA or the OSSE data . We combine all existing X-ray and hard X-ray observations and present the orbital modulation in the luminosity and photon index for the entire orbit . The results are in agreement with predictions based on a synchrotron emission model from relativistic particles in a shocked pulsar wind interacting with the gaseous outflow from the Be star .