Spin-polarized band structure of the three-dimensional quantum spin Hall insulator Bi1-xSbx (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, Γ¯ and M¯ , of the (111) surface Brillouin zone, a spin-up band ( Σ1' 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 -1 , which agrees with the theoretical prediction based on first-principles calculations.