The binary Re1-xMox alloys, known to cover the full range of solid solutions, were successfully synthesized and their crystal structures and physical properties investigated via powder x-ray diffraction, electrical resistivity, magnetic susceptibility, and heat capacity. By varying the Re/Mo ratio, we explore the full Re1-xMox binary phase diagram, in all its four different solid phases: hcp-Mg (P6(3)/mmc), alpha-Mn (1 (4) over bar 3m) beta-CrFe (P4(2)/mnm), and bcc-W (Im (3) over barm), of which the second is noncentrosymmetric with the rest being centrosymmetric. All Re1-xMox alloys are superconductors, whose critical temperatures exhibit a peculiar phase diagram, characterized by three different superconducting regions. In most alloys, the T-c is almost an order of magnitude higher than in pure Re and Mo. Low-temperature electronic specific-heat data evidence a fully gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling across the series. Considering that several alpha-Mn-type ReT alloys (T = transition metal) show time-reversal symmetry breaking (TRSB) in the superconducting state, while TRS is preserved in the isostructural Mg10Ir19B16 or Nb0.5Os0.5, the Re1-xMox alloys represent another suitable system for studying the interplay of space-inversion, gauge, and time-reversal symmetries in future experiments expected to probe TRSB in the ReT family.