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 \sim 6 \times 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 .