  The photon mass, $m_\gamma$, can in principle be constrained using
measurements of the dispersion measures (DMs) of fast radio bursts (FRBs), once
the FRB redshifts are known. The DM of the repeating FRB 121102 is known to $<
1$\%, a host galaxy has now been identified with high confidence,and its
redshift, $z$, has now been determined with high accuracy: $z = 0.19273(8)$.
Taking into account the plasma contributions to the DM from the Intergalactic
medium (IGM) and the Milky Way, we use the data on FRB 121102 to derive the
constraint $m_\gamma \lesssim 2.2 \times 10^{-14}$ eV c$^{-2}$ ($3.9 \times
10^{-50}$ kg). Since the plasma and photon mass contributions to DMs have
different redshift dependences, they could in principle be distinguished by
measurements of more FRB redshifts, enabling the sensitivity to $m_\gamma$ to
be improved.
