  We derive constraints on the time-averaged event rate of neutron star-black
hole (NS-BH) mergers by using estimates of the population-integrated production
of heavy rapid neutron-capture (r-process) elements with nuclear mass numbers A
> 140 by such events in comparison to the Galactic repository of these chemical
species. Our estimates are based on relativistic hydrodynamical simulations
convolved with theoretical predictions of the binary population. This allows us
to determine a strict upper limit of the average NS-BH merger rate of ~6*10^-5
per year. We quantify the uncertainties of this estimate to be within factors
of a few mostly because of the unknown BH spin distribution of such systems,
the uncertain equation of state of NS matter, and possible errors in the
Galactic content of r-process material. Our approach implies a correlation
between the merger rates of NS-BH binaries and of double NS systems.
Predictions of the detection rate of gravitational-wave signals from such
compact-object binaries by Advanced LIGO and Advanced Virgo on the optimistic
side are incompatible with the constraints set by our analysis.
