By comparing the observed orbital decay of the binary pulsars PSR B1913+16 and PSR B1534+12 to that predicted by general relativity due to gravitational-wave emission, we are able to bound the mass of the graviton to be less than 7.6 × 10⁻²⁰ eV/c² at 90 confidence. This is the first such bound to be derived from dynamic gravitational fields. It is approximately two orders of magnitude weaker than the static-field bound from solar system observations, and will improve with further observations.