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The cross section of the process $e^+e^-\to\pi^+\pi^-\pi^0$ is measured witha precision of 1.6% to 25% in the energy range between $0.7$ and 3.0 GeV usingthe Initial State Radiation method. A data set with an integrated luminosity of$2.93$fb$^{-1}$ taken at the center-of-mass energy of $\sqrt{s}=3.773$GeV withthe BESIII detector at the BEPCII collider is used. The product branchingfractions for $\omega$, $\phi$, $\omega(1420)$, and $\omega(1650)$ are measuredto be $\mathcal{B}(\omega\to e^+e^-) \times \mathcal{B}(\omega \to\pi^+\pi^-\pi^0)=(6.94\pm0.08\pm0.16) \times 10^{-5}$, $\mathcal{B}(\phi\toe^+e^-) \times \mathcal{B}(\phi\to\pi^+\pi^-\pi^0) = (4.20\pm0.08\pm0.19)\times 10^{-5}$, $\mathcal{B}(\omega(1420)\to e^+e^-) \times\mathcal{B}(\omega(1420) \to \pi^+\pi^-\pi^0) = (0.84\pm0.09\pm0.09) \times10^{-6}$, and $\mathcal{B}(\omega(1650) \to e^+e^) \times\mathcal{B}(\omega(1650)\to \pi^+\pi^-\pi^0) =(1.14\pm0.15\pm0.15)\times10^{-6}$, respectively. The branching fraction$\mathcal{B}(J/\psi\to \pi^+\pi^-\pi^0)$ is measured to be $(2.188 \pm 0.024\pm 0.024 \pm0.040 (\Gamma_{ee}^{J/\psi}))\%$, where $\Gamma_{ee}^{J/\psi}$ isthe dileptonic width of $J/\psi$. The first errors are of statistical, thesecond and third ones of systematic nature.

Measurement of the $e^+e^-\to\pi^+\pi^-\pi^0$ cross section from 0.7 GeV to 3.0 GeV via initial-state radiation