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, but independent verification was not possible. Such verification can in principle be obtained by determining the orbital inclination in a binary pulsar system using only classical geometrical constraints. This would permit a measurement of the expected retardation of the pulse signal arising from the general relativistic curvature of space-time in the vicinity of the companion object (the 'Shapiro delay'). Here we report high-precision radio observations of the binary millisecond pulsar PSR J0437-4715, which establish the three-dimensional structure of its orbit. We see the Shapiro delay predicted by general relativity, and we determine the mass of the neutron star and its white dwarf companion. The determination of such masses is necessary in order to understand the origin and evolution of neutron stars.