Since the \ensuremath{\beta} decay of $^{187}\mathrm{Re}$ is first-forbidden unique, the ratio \ensuremath{\rho} of bound-state to continuum \ensuremath{\beta} decay of $^{187}\mathrm{Re}$ reduces to a problem in atomic physics. A knowledge of \ensuremath{\rho} is required if one is to connect the experimentally determined value of the rate of continuum decay to a value of the half-life ${T}_{1/2}$ for the \ensuremath{\beta} decay of $^{187}\mathrm{Re}$. The value of ${T}_{1/2}$ plays a significant role in the use of the geochemically determined ratio of $^{187}\mathrm{Os}$ to $^{187}\mathrm{Re}$ in the estimation of the age of our galaxy. Our results for \ensuremath{\rho}, based on the Hartree-Fock-Dirac single-configuration and the multiconfiguration Dirac-Fock approximations strengthen the conclusion of Williams, Fowler, and Koonin [Astrophys. J. 281, 363 (1984)] based on a modified Thomas-Fermi model, that \ensuremath{\rho} is only of the order of 1%. We also include a few values of the rate for a process first considered by Perrone [Ph.D. thesis, Rice University (1971)], the bound-state \ensuremath{\beta} decay of a bare $^{187}\mathrm{Re}$ nucleus. (Since the mass of the $^{187}\mathrm{Re}$ nucleus is less than the sum of the masses of its daughter nucleus $^{187}\mathrm{Os}$ and of an electron, continuum decay of a bare $^{187}\mathrm{Re}$ nucleus is forbidden; neutral $^{187}\mathrm{Re}$ can \ensuremath{\beta} decay only because of the greater binding energy of the electrons of $^{187}\mathrm{Os}$ than of $^{187}$Re.)