High-resolution spin-and angle-resolved photoemission spectroscopy measurements were performed on the three-dimensional topological insulator Bi2Te2.4Se0.6, which is characterized by enhanced thermoelectric properties. The Fermi level position is found to be located in the bulk energy gap independent of temperature and it is stable over a long time. Spin textures in the Dirac-cone state at energies above and below the Dirac point as well as in the Rashba-type valence band surface state are observed in agreement with theoretical prediction. The calculations of the surface electronic structure demonstrate that the fractional stoichiometry induced disorder within the Te/Se sublattice does not influence the Dirac-cone state dispersion. In spite of relatively high resistivity, temperature dependence of conductivity shows a weak metallic behavior that could explain the effective thermoelectric properties of the Bi2Te2.4Se0.6 compound with the in-plane Seebeck coefficient reaching -330 mu V/K at room temperature.