  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.
