Spin-polarized band structure of the three-dimensional quantum spin Hall insulator ${\text{Bi}}_{1\ensuremath{-}x}{\text{Sb}}_{x}$ $(x=0.12--0.13)$ was fully elucidated by spin-polarized angle-resolved photoemission spectroscopy using a high-yield spin polarimeter equipped with a high-resolution electron spectrometer. Between the two time-reversal-invariant points, $\overline{\ensuremath{\Gamma}}$ and $\overline{\text{M}}$, of the (111) surface Brillouin zone, a spin-up band (${\ensuremath{\Sigma}}_{1}^{\ensuremath{'}}$ band) was found to cross the Fermi energy only once, providing unambiguous evidence for the strong topological insulator phase. The observed spin-polarized band dispersions determine the ``mirror chirality'' to be $\ensuremath{-}1$, which agrees with the theoretical prediction based on first-principles calculations.