CUPID-0 is the first pilot experiment of CUPID, a next-generation project for the measurement of neutrinoless double beta decay (0νDBD) 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×yr. In this Letter we present the phase-I results in the search for 0νDBD. We demonstrate that the technology implemented by CUPID-0 allows us to reach the lowest background for calorimetric experiments: (3.5_{-0.9}^{+1.0})×10^{-3} counts/(keV kg yr). Monitoring 3.88×10^{25} ^{82}Se nuclei×yr we reach a 90% credible interval median sensitivity of T_{1/2}^{0ν}>5.0×10^{24} yr and set the most stringent limit on the half-life of ^{82}Se 0νDBD: T_{1/2}^{0ν}>3.5×10^{24} yr (90% credible interval), corresponding to m_{ββ}<(311-638) meV depending on the nuclear matrix element calculations.