We present relativistic analyses of 9257 measurements of times-of-arrival from the first binary pulsar, PSR B1913+16, acquired over the last thirty-five years. The determination of the Keplerian orbital elements plus two relativistic terms completely characterizes the binary system, aside from an unknown rotation about the line of sight; leading to a determination of the masses of the pulsar and its companion: 1.438 $pm$ 0.001 solar masses and 1.390 $pm$ 0.001 solar masses, respectively. In addition, the complete system characterization allows the creation of tests of relativistic gravitation by comparing measured and predicted sizes of various relativistic phenomena. We find that the ratio of observed orbital period decrease due to gravitational wave damping (corrected by a kinematic term) to the general relativistic prediction, is 0.9983 pm 0.0016; thereby confirming the existence and strength of gravitational radiation as predicted by general relativity. For the first time in this system, we have also successfully measured the two parameters characterizing the Shapiro gravitational propagation delay, and find that their values are consistent with general relativistic predictions. We have also measured for the first time in any system the relativistic shape correction to the elliptical orbit, $delta_{theta}$,although its intrinsic value is obscured by currently unquantified pulsar emission beam aberration. We have also marginally measured the time derivative of the projected semimajor axis, which, when improved in combination with beam aberration modelling from geodetic precession observations, should ultimately constrain the pulsars moment of inertia.
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