Multigap superconductivity, emerging in metals with several bands crossing the Fermi level, favors exotic superconducting orders that have no equivalent in a single-band counterpart. In this context, it is important to search for new materials with well-established two (or more) gaps having distinctly different sizes. In this work, we confirm previous statements and present new evidence to support the claim that Mo-Re alloy with a comparable concentration of the components is a two-band/two-gap superconductor. The differential conductance spectra obtained in point-contact experiments demonstrate the presence of a bosonic, undamped collective mode and its harmonics associated with the superconducting state. Following previous works on MgB2, we have identified these features as manifestations of the so-called Leggett mode arising due to relative phase fluctuations between two superconducting order parameters.