  We report the first observation of $e^+e^-$ annihilations into states of
positive $C$-parity, namely $\rho^0 \rho^0$ and $\phi\rho^0$. The two states
are observed in the $\pip\pim\pip\pim$ and $K^+K^-\pip\pim$ final states,
respectively, in a data sample of 225~$fb^{-1}$ collected by the
$BaBar$~experiment at the $PEP-II$ $e^+e^-$ storage rings at energies near
$\sqrt{s} ~=$ 10.58~$GeV$. The distributions of $\cos\theta^*$, where
$\theta^*$ is the center-of-mass polar angle of the $\phi$ meson or the forward
$\rho^0$ meson, suggest production by two-virtual-photon annihilation. We
measure cross sections within the range $|\cos\theta^*| ~<~ 0.8$ of
$\sigma(\e^+e^- \to \rho^0 \rho^0) ~=~$$20.7 \pm 0.7 (\mathrm{stat})\pm
2.7(\mathrm{syst}) ~\mathrm{fb}$ and $\sigma(\e^+e^- \to \phi \rho^0)
\~=~$$5.7\pm 0.5(\mathrm{stat}) \pm 0.8(\mathrm{syst}) ~\mathrm{fb}$.
