Final result of CUPID-0 phase-I in the search for the $^{82}$Se Neutrinoless Double Beta Decay

CUPID-0 is the first pilot experiment of CUPID, a next-generation project for the measurement of neutrinoless double beta decay (0$\nu$DBD) with scintillating bolometers. The detector, consisting of 24 enriched and 2 natural ZnSe crystals, has been taking data at Laboratori Nazionali del Gran Sasso from June 2017 to December 2018, collecting a $^{82}$Se exposure of 5.29 kg$\times$yr. In this paper we present the phase-I results in the search for 0$\nu$DBD. We demonstrate that the technology implemented by CUPID-0 allows us to reach the lowest background for calorimetric experiments: $(3.5^{+1.0}_{-0.9})\times10^{-3}$ counts/(keV kg yr). Monitoring 3.88$\times$10$^{25}$ $^{82}$Se nuclei$\times$yr we reach a 90% credible interval median sensitivity of $\rm{T}^{0\nu}_{1/2}>5.0\times10^{24} \rm{yr}$ and set the most stringent limit on the half-life of $^{82}$Se 0$\nu$DBD : $\rm{T}^{0\nu}_{1/2}>3.5\times10^{24} \rm{yr}$ (90% credible interval), corresponding to m$_{\beta\beta} <$ (311-638) meV depending on the nuclear matrix element calculations.