This thesis reports on the recent results of a continuing, high-precision pulsar timing project, currently focused on the nearby, binary millisecond pulsar, PSR J0437_4715. Pulse arrival time analysis has yielded a remarkable series of constraints on the physical parameters of this system and evidence for the distortion of space-time as predicted by the General Theory of Relativity. Owing to the proximity of the PSR J0437_4715 system, relative changes in the positions of the Earth and pulsar result in both annual and secular evolution of the line of sight to the pulsar. Although the changes are miniscule, the effects on the projected orbital parameters are detectable in our data at a high level of significance, necessitating the implementation of an improved timing model. In addition to producing estimates of astrometric parameters with unparalleled precision, the study has also yielded the first three-dimensional orbital geometry of a binary pulsar. This achievement includes the first classical determination of the orbital inclination, thereby providing the unique opportunity to verify the shape of the Shapiro delay and independently confirm a general relativistic prediction. With a current post-fit arrival time residual RMS of 130 ns over four years, the unrivaled quality of the timing data presented herein may eventually contribute to the most stringent limit on the energy density of the proposed stochastic gravitational wave background. Continuing the quest for even greater timing precision, a detailed study of the polarimetry of PSR J0437_4715 was undertaken. This effort culminated in the development of a new, phase-coherent technique for calibrating the instrumental response of the observing system. Observations were conducted at the Parkes 64-m radio telescope in New South Wales, Australia, using baseband recorder technologies developed at York University, Toronto, and at the California Institute of Technology. Data were processed off-line at Swinburne University using a beowulf-style cluster of high-performance workstations and custom software developed by the candidate as part of this thesis.