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.