We report on the measurement of the two-neutrino double-$beta$ decay of $^{82}$Se performed for the first time with cryogenic calorimeters, in the framework of the CUPID-0~experiment. With an exposure of 9.95 kg$times$yr of Zn$^{82}$Se, we determine the two-neutrino double-$beta$ decay half-life of $^{82}$Se with an unprecedented precision level, $T_{1/2}^{2 u} = [8.60 pm 0.03 textrm{(stat.)}~^{+0.17}_{-0.10} textrm{(syst.)}] times 10^{19}~textrm{yr}$. The very high signal-to-background ratio, along with the detailed reconstruction of the background sources allowed us to identify the single state dominance as the underlying mechanism of such process, demonstrating that the higher state dominance hypothesis is disfavored at the level of 5.5 $sigma$.
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