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~fb$^{^-1}$ of integrated luminosity, collected at a center-of-mass energy at or near the $\Upsilon (4S)$ resonance with the \belletwo 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 $a^{3\pi}_{\mu} = [48.91 \pm 0.23~(\mathrm{stat}) \pm 1.07~(\mathrm{syst})] \times 10^{-10}$ 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.