  The first observation of the $B^0_s\!\to D^{*+}D^{*-}$ decay and the
measurement of its branching ratio relative to the $B^0\!\to D^{*+}D^{*-}$
decay are presented. The data sample used corresponds to an integrated
luminosity of $9\,\text{fb}^{-1}$ of proton-proton collisions recorded by the
LHCb experiment at centre-of-mass energies of 7, 8 and $13\,\text{TeV}$ between
2011 and 2018. The decay is observed with more than $10$ standard deviations
and the time-integrated ratio of branching fractions is determined to be
\begin{align*}
  \frac{\mathcal{B}(B^0_s\!\to D^{*+}D^{*-})}{\mathcal{B}(B^0\!\to
D^{*+}D^{*-})} = 0.269 \pm 0.032 \pm 0.011 \pm 0.008\, , \end{align*} where the
first uncertainty is statistical, the second systematic and the third due to
the uncertainty of the fragmentation fraction ratio $f_s/f_d$. The $B^0_s\!\to
D^{*+}D^{*-}$ branching fraction is calculated to be \begin{align*}
  \mathcal{B}(B^0_s\!\to D^{*+}D^{*-}) = (2.15 \pm 0.26 \pm 0.09 \pm 0.06 \pm
0.16)\times 10^{-4} \,, \end{align*} where the fourth uncertainty is due to the
$B^0\!\to D^{*+}D^{*-}$branching fraction. These results are calculated using
the average $B^0_s$ meson lifetime in simulation. Correction factors are
reported for scenarios where either a purely heavy or a purely light $B^0_s$
eigenstate is considered.
