We present a determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ from the decay $B\to D\ellν_\ell$ using a $365~\mathrm{fb}^{-1}$$e^+e^-\toΥ(4S)\to B\bar B$ data sample recorded by the Belle II experiment at the SuperKEKB collider. The semileptonic decay of one $B$ meson is reconstructed in the modes $B^0\to D^-(\to K^+π^-π^-)\ell^+ν_\ell$ and $B^+\to \bar D^0(\to K^+π^-)\ell^+ν_\ell$, where $\ell$ denotes either an electron or a muon. Charge conjugation is implied. The second $B$ meson in the $Υ(4S)$ event is not reconstructed explicitly. Using an inclusive reconstruction of the unobserved neutrino momentum, we determine the recoil variable $w=v_B\cdot v_D$, where $v_B$ and $v_D$ are the 4-velocities of the $B$ and $D$ mesons. We measure the total decay branching fractions to be $\mathcal{B}(B^0\to D^-\ell^+ν_\ell)=(2.06 \pm 0.05\,(\mathrm{stat.}) \pm 0.10\,(\mathrm{sys.}))\%$ and $\mathcal{B}(B^+\to\bar D^0\ell^+ν_\ell)=(2.31 \pm 0.04\,(\mathrm{stat.}) \pm 0.09\,(\mathrm{sys.}))\%$. We probe lepton flavor universality by measuring $\mathcal{B}(B\to Deν_e)/\mathcal{B}(B\to Dμν_μ)=1.020 \pm 0.020\,(\mathrm{stat.})\pm 0.022\,(\mathrm{sys.})$. Fitting the partial decay branching fraction as a function of $w$ and using the average of lattice QCD calculations of the $B\to D$ form factor, we obtain $ |V_{cb}|=(39.2\pm 0.4\,(\mathrm{stat.}) \pm 0.6\,(\mathrm{sys.}) \pm 0.5\,(\mathrm{th.})$.
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