  A search for the $B^0_s \!\rightarrow D^{*\pm} D^\mp$ decay is performed
using proton-proton collision data at centre-of-mass energies of $7$, $8$ and
$13\,\text{TeV}$ collected by the LHCb experiment, corresponding to an
integrated luminosity of $9\,\text{fb}^{-1}$. The decay is observed with a high
significance and its branching fraction relative to the $B^0 \!\rightarrow
D^{*\pm} D^\mp$ decay is measured to be \begin{align*} \frac{\mathcal{B}(B_s^0
\rightarrow D^{\ast \pm} D^{\mp}) }{\mathcal{B}(B^0 \rightarrow D^{\ast \pm}
D^{\mp}) } = 0.137 \pm 0.017 \pm 0.002 \pm 0.006 \,, \end{align*} where the
first uncertainty is statistical, the second systematic and the third is due to
the uncertainty on the ratio of the $B_s^0$ and $B^0$ hadronisation fractions.
