The branching ratio of the electron-capture decay of $^{81}\mathrm{Kr}^{\mathrm{m}}$ (13 s) has been measured in order to obtain the rate of the neutrino capture reaction $^{81}\mathrm{Br}$(\ensuremath{\nu},${\mathrm{re}}^{\ensuremath{-}}$${)}^{81}$Kr(190.6 keV). This quantity is needed to determine the properties of an $^{81}\mathrm{Br}$ solar neutrino detector. A branching ratio of (3.14\ifmmode\pm\else\textpm\fi{}0.58)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ was obtained, leading to a logft of ${5.10}_{\mathrm{\ensuremath{-}}0.07}^{+0.06}$ for this transition.