Binary pulsars provide an excellent system for testing general relativity because of their intrinsic rotational stability and the precision with which radio observations can be used to determine their orbital dynamics. Measurements of the rate of orbital decay of two pulsars have been shown to be consistent with the emission of gravitational waves as predicted by general relativity, providing the most convincing evidence for the self-consistency of the theory to date. However, independent verification of the orbital geometry in these systems was not possible. Such verification may be obtained by determining the orientation of a binary pulsar system using only classical geometric constraints, permitting an independent prediction of general relativistic effects. Here we report high-precision timing of the nearby binary millisecond pulsar PSR J0437-4715, which establish the three-dimensional structure of its orbit. We see the expected retardation of the pulse signal arising from the curvature of space-time in the vicinity of the companion object (the `Shapiro delay'), and we determine the mass of the pulsar and its white dwarf companion. Such mass determinations contribute to our understanding of the origin and evolution of neutron stars. ; Comment: 5 pages, 2 figures