  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, $T_{1/2}^{2\nu} = [8.60 \pm
0.03 \textrm{(stat.)}~^{+0.17}_{-0.10} \textrm{(syst.)}] \times
10^{19}~\textrm{yr}$. 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 $\sigma$.
