Analysis of the quantum oscillations of magnetoresistance (the Shubnikov-de Haas effect) in Bi1 - x Sb x alloys with an antimony content in the range 0.255 < x < 0.260 has revealed a Lifshitz electronic-topological transition, which quite possibly can be explained in terms of the existence of a saddle point in the energy spectrum of these compositions. Such a peculiarity comes into existence when the direct band gap at the L point of the Brillouin zone in the semiconductor region of the compounds with x > 0.04 becomes negative. This compel one to revise essentially all earlier calculations based on the previously obtained values of the band parameters. In order to check the agreement between the new values of the band parameters and the data on the density of states obtained from measurements of the thermopower in the classical limit of strong magnetic fields, theoretical calculations of the charge carrier concentration n and the density of states at the Fermi level ρ( E F) have been performed for the case of negative values of the direct band gap at the L point E gL. The calculations of the parameters n and ρ( E F) have demonstrated that the change in E gL and the corresponding correction of the band parameters ensure good agreement with the experimental data. According to these calculations, one electronic-topological transition occurs at an antimony content x ∼ 0.165, when a saddle point appears in the energy spectrum. The second transition is associated with the transformation of the six ellipsoids of the Fermi surface into three dumbbell-like figures at antimony concentrations in the range 0.255 < x < 0.260.