  The cyclotron frequency ratio of $^{187}\mathrm{Os}^{29+}$ to
$^{187}\mathrm{Re}^{29+}$ ions was measured with the Penning-trap mass
spectrometer PENTATRAP. The achieved result of $R=1.000\:000\:013\:882(5)$ is
to date the most precise such measurement performed on ions. Furthermore, the
total binding-energy difference of the 29 missing electrons in Re and Os was
calculated by relativistic multiconfiguration methods, yielding the value of
$\Delta E = 53.5(10)$ eV. Finally, using the achieved results, the mass
difference between neutral $^{187}$Re and $^{187}$Os, i.e., the $Q$ value of
the $\beta^-$ decay of $^{187}$Re, is determined to be 2470.9(13) eV.
