  We study the process $e^+e^- \to \pi^+\pi^-\eta \gamma$, where the photon is
radiated from the initial state. About 8000 fully reconstructed events of this
process are selected from the BaBar data sample with an integrated luminosity
of 469~fb$^{-1}$. Using the $\pi^+\pi^-\eta$ invariant mass spectrum we measure
the $e^+e^- \to \pi^+\pi^-\eta$ cross section in the $e^+e^-$ center-of-mass
energy range from 1.15 to 3.5 GeV. The cross section is well described by the
Vector-Meson Dominance model with four $\rho$-like states. We observe $49\pm9$
events of the $J/\psi$ decay to $\pi^+\pi^-\eta$, and measure the product
$\Gamma_{J/\Psi \to e^+e^-} \text{B}_{J/\Psi \to \pi^+\pi^-\eta} = 2.34 \pm
0.43_{\text{stat}} \pm 0.16_{\text{syst}}$~eV.
