  The ratio of branching fractions $R(D^{*}) = \mathcal{B}(\overline{B}
\rightarrow D^{*} \tau^{-} \overline{\nu}_{\tau})$/$\mathcal{B} (\overline{B}
\rightarrow D^{*} \ell^{-} \overline{\nu}_{\ell})$, where $\ell$ is an electron
or muon, is measured using a Belle~II data sample with an integrated luminosity
of $189~\mathrm{fb}^{-1}$ at the SuperKEKB asymmetric-energy $e^{+} e^{-}$
collider. Data is collected at the $\Upsilon(\mathrm{4S})$ resonance, and one
$B$ meson in the $\Upsilon(\mathrm{4S})\rightarrow B\overline{B}$ decay is
fully reconstructed in hadronic decay modes. The accompanying signal $B$ meson
is reconstructed as $\overline{B}\rightarrow D^{*}
\tau^{-}\overline{\nu}_{\tau}$ using leptonic $\tau$ decays. The normalization
decay, $\overline{B}\rightarrow D^{*} \ell^{-} \overline{\nu}_{\ell}$, where
$\ell$ is an electron or muon, produces the same observable final state
particles. The ratio of branching fractions is extracted in a simultaneous fit
to two signal-discriminating variables in both channels and yields $R(D^{*}) =
0.262~_{-0.039}^{+0.041}(\mathrm{stat})~_{-0.032}^{+0.035}(\mathrm{syst})$.
This result is consistent with the current world average and with standard
model predictions.
