  We report a measurement of the $e^+e^- \to \pi^+\pi^-\pi^0$ cross section in
the energy range from 0.62 to 3.50 GeV using an initial-state radiation
technique. We use an $e^+e^-$ data sample corresponding to 191 $\text{fb}^{-1}$
of integrated luminosity, collected at a center-of-mass energy at or near the
$\Upsilon{(4S)}$ resonance with the Belle II detector at the SuperKEKB
collider. Signal yields are extracted by fitting the two-photon mass
distribution in $e^+e^- \to \pi^+\pi^-\pi^0\gamma$ events, which involve a
$\pi^0 \to \gamma\gamma$ decay and an energetic photon radiated from the
initial state. Signal efficiency corrections with an accuracy of 1.6% are
obtained from several control data samples. The uncertainty on the cross
section at the $\omega$ and $\phi$ resonances is dominated by the systematic
uncertainty of 2.2%. The resulting cross sections in the 0.62-1.80 GeV energy
range yield $ a_\mu^{3\pi} = [48.91 \pm 0.23~(\mathrm{stat}) \pm
1.07~(\mathrm{syst})] \times 10^{-10} $ for the leading-order hadronic vacuum
polarization contribution to the muon anomalous magnetic moment. This result
differs by $2.5$ standard deviations from the most precise current
determination.
