  In Einstein's general relativity, gravity is mediated by a massless spin-2
metric field, and its extension to include a mass for the graviton has profound
implication for gravitation and cosmology. In 2002, Finn and Sutton used the
gravitational-wave (GW) back-reaction in binary pulsars, and provided the first
bound on the mass of graviton. Here we provide an improved analysis using 9
well-timed binary pulsars with a phenomenological treatment. First, individual
mass bounds from each pulsar are obtained in the frequentist approach with the
help of an ordering principle. The best upper limit on the graviton mass,
$m_{g}<3.5\times10^{-20} \, {\rm eV}/c^{2}$ (90% C.L.), comes from the
Hulse-Taylor pulsar PSR B1913+16. Then, we combine individual pulsars using the
Bayesian theorem, and get $m_{g}<5.2\times10^{-21} \, {\rm eV}/c^{2}$ (90%
C.L.) with a uniform prior for $\ln m_g$. This limit improves the Finn-Sutton
limit by a factor of more than 10. Though it is not as tight as those from GWs
and the Solar System, it provides an independent and complementary bound from a
dynamic regime.
