CUPID-0 is the first large array of scintillating Zn 82 Se cryogenic calorimeters (bolometers) implementing particle identification for the search of the neutrinoless double beta decay (0 ν β β ). The detector consists of 24 enriched Zn 82 Se bolometers for a total 82 Se mass of 5.28 kg and it has been taking data in the underground LNGS (Italy) since March 2017. In this article we show how the dual read-out provides a powerful tool for the α particles rejection. The simultaneous use of the heat and light information allows us to reduce the background down to (3.2 − 1.1 + 1.3 )×10 − 3 counts/(keV kg year), an unprecedented level for cryogenic calorimeters. In a total exposure of 5.46 kg year Zn 82 Se we set the most stringent limit on the 0 ν β β decay 82 Se half-life T 1 / 2 0 ν > 4.0 × 10 24 year at 90% C.I.