Thin films of ${\rm Cu_2Mo_6S_8}$ have been in-situ deposited by pulsed laser deposition on R-plane ${\rm Al_2O_3}$ on which an epitaxial growth has been achieved for the first time, with the $(100)_R$ orientation, as shown by X-ray diffraction. The superconducting transition temperature has been determined from DC resistive and AC susceptibility measurements. The critical temperature Tc is typically between 9.5 and 10.3 K and the transitions are narrow: the full width at half maximum of the inductive signal $\chi ''$ ranges between 0.1 and 0.5 K. The critical current densities have been obtained by standard four probe electrical measurements: in zero field a value larger than $10^8\; {\rm A}/{\rm m}^2$ has been measured at only 1 K from the end of the transition, and values typically larger than $10^7\;{\rm A}/{\rm m}^2$ at 4.2 K in an applied magnetic field of 7 T have been observed. The upper critical magnetic field, Bc2, has been determined from the critical current measurements performed at 4.2 K in an applied magnetic field, for two samples. The experimental values of 11 and 12.5 T at 4.2 K are close to those reported on bulk samples (12.5 T). For the first time to our knowledge a surface resistance $(R_{\rm s})$ measurement has been carried out on a Chevrel phase-type compound: a value of 4.5 mΩ at 10 GHz and 4.2 K has been obtained.