  Photon is the fundamental quantum of electromagnetic fields, whose mass,
$m_{\gamma}$, should be strictly zero in Maxwell's theory. But not all theories
adopt this hypothesis. If the rest mass of the photon is not zero, there will
be an additional time delay between photons of different frequencies after they
travel through a fixed distance. By analyzing the time delay, we can measure or
constrain the photon mass. Fast radio bursts (FRBs) -- transient radio bursts
characterized by millisecond duration and cosmological propagation -- are
excellent astrophysical laboratories to constrain $m_{\gamma}$. In this work we
use a catalog of 129 FRBs in a Bayesian framework to constrain $m_{\gamma}$. As
a result, we obtain a new bound on the photon mass, $m_{\gamma} \leq 3.1\times
10^{-51}\rm\,kg\simeq 1.7 \times 10^{-15}\,eV/c^2$ ($m_{\gamma} \leq 3.9\times
10^{-51}\rm\,kg \simeq 2.2 \times 10^{-15}\,eV/c^2$) at the $68\%$ $(95\%$)
confidence level. The result represents the best limit purely from kinematic
analysis of light propagation. The bound on the photon mass will be tighter in
the near future with increment in the number of FRBs, more accurate measurement
of the redshift for FRBs, and refinement in the knowledge about the origin of
dispersion measures (DMs).
