  We measure the branching fractions for the decays $\bar{B}^0 \to D^{*+} e^{-}
\bar{\nu}_e$ and $\bar{B}^0 \to D^{*+} \mu^{-} \bar{\nu}_\mu$ using $8.70 \pm
0.09~{\rm fb}^{-1}$ of data collected by the Belle II experiment at the
SuperKEKB asymmetric-energy $e^+ e^-$ collider. Candidate signal decays are
reconstructed with the subsequent decays $D^{*+}\to D^0 \pi^+$ and $D^0\to
K^-\pi^+$. We obtain the results ${\cal B}(\bar{B}^0 \to D^{*+} e^{-}
\bar{\nu}_e)$ = $(4.55\pm0.14(\mathrm{stat})\pm0.35 (\mathrm{syst})) \%$ and
${\cal B}(\bar{B}^0 \to D^{*+} \mu^{-} \bar{\nu}_\mu)$ = $(4.84\pm
0.13(\mathrm{stat})\pm0.37(\mathrm{syst})) \%$, in agreement with the world
averages. The measurements serve to validate the full chain of detector
operation and calibration, data collection and processing, and production of
physics results in the case of semileptonic $B$-meson decays.
