  Fast radio bursts (FRBs) are very short and bright transients visible over
extragalactic distances. The radio pulse undergoes dispersion caused by free
electrons along the line of sight, most of which are associated with the
large-scale structure (LSS). The total dispersion measure therefore increases
with the line of sight and provides a distance estimate to the source. We
present the first measurement of the Hubble constant using the dispersion
measure -- redshift relation of FRBs with identified host counterpart and
corresponding redshift information. A sample of nine currently available FRBs
yields a constraint of $H_0 = 62.3 \pm 9.1 \,\rm{km}
\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$, accounting for uncertainty stemming from the
LSS, host halo and Milky Way contributions to the observed dispersion measure.
The main current limitation is statistical, and we estimate that a few hundred
events with corresponding redshifts are sufficient for a per cent measurement
of $H_0$. This is a number well within reach of ongoing FRB searches. We
perform a forecast using a realistic mock sample to demonstrate that a
high-precision measurement of the expansion rate is possible without relying on
other cosmological probes. FRBs can therefore arbitrate the current tension
between early and late time measurements of $H_0$ in the near future.
