Cryogenic calorimeters are currently used by experiments searching for neutrino-less double beta decay (0$\nu\beta\beta$). The sensitivity of these experiments is limited by the background coming from $\alpha$ radioactive contaminations. This background can be rejected exploiting the simultaneous read-out of light and heat in scintillating cryogenic calorimeters, because both the amplitude and the time development of the light signal depend on the nature of the particle interacting within. In this paper we present the CUPID-0 detector which represents the first demonstrator of this technique. Exploiting an array of 26 Zn$^{82}$Se scintillating crystals operated as calorimeters and monitored by 31 cryogenic light detectors, CUPID-0 demonstrated the capability to completely reject the $\alpha$ background, paving the way for a next generation experiment searching for 0$\nu\beta\beta$.