The pulsar/massive star binary system PSR B1259-63/LS 2883 is one of the best-studied gamma-ray binaries, a class of systems whose bright gamma-ray flaring can provide important insights into high-energy physics. Using the Australian Long Baseline Array, we have conducted very long baseline interferometric observations of PSR B1259-63 over 4.4 years, fully sampling the 3.4-year orbital period. From our measured parallax of 0.38 ± 0.05 maswe use a Bayesian approach to infer a distance of 2.6+0.4 -0.3 kpc. We find that the binary orbit is viewed at an angle of 154 ± 3° to the line of sight, implying that the pulsar moves clockwise around its orbit as viewed on the sky. Taking our findings together with previous results from pulsar timing observations, all seven orbital elements for the system are now fully determined. We use our measurement of the inclination angle to constrain the mass of the stellar companion to lie in the range 15-31M⊙. Our measured distance and proper motion are consistent with the system having originated in the Cen OB1 association and receiving a modest natal kick, causing it to have moved ~8 pc from its birthplace over the past ~3 × 105 years. The orientation of the orbit on the plane of the sky matches the direction of motion of the X-ray synchrotron-emitting knot observed by the Chandra X-ray Observatory to be moving away from the system.