  The cross section of the process $e^+e^-\to\pi^+\pi^-\pi^0$ is measured with
a precision of 1.6% to 25% in the energy range between $0.7$ and 3.0 GeV using
the 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 with
the BESIII detector at the BEPCII collider is used. The product branching
fractions for $\omega$, $\phi$, $\omega(1420)$, and $\omega(1650)$ are measured
to be $\mathcal{B}(\omega\to e^+e^-) \times \mathcal{B}(\omega \to
\pi^+\pi^-\pi^0)=(6.94\pm0.08\pm0.16) \times 10^{-5}$, $\mathcal{B}(\phi\to
e^+e^-) \times \mathcal{B}(\phi\to\pi^+\pi^-\pi^0) = (4.20\pm0.08\pm0.19)
\times 10^{-5}$, $\mathcal{B}(\omega(1420)\to e^+e^-) \times
\mathcal{B}(\omega(1420) \to \pi^+\pi^-\pi^0) = (0.84\pm0.09\pm0.09) \times
10^{-6}$, 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 $\Gamma_{ee}^{J/\psi}$ is
the dileptonic width of $J/\psi$. The first errors are of statistical, the
second and third ones of systematic nature.
