  Direct CP violation in the hadronic decays $\Bar{B}^{0} \to \pi^{+}\pi^{-}
\pi^{0}$ is investigated near the peak of the $\rho^{0}$ taking into account
the effect of $\rho - \omega$ mixing. Branching ratios for processes $B^{\pm,0}
\to \rho^{\pm,0}\pi^{\pm,0}$ and $B^{-} \to \omega \pi^{-}$ are calculated as
well. We find that the CP violating asymmetry is strongly dependent on the CKM
matrix elements. For a fixed $N_{c}^{eff}$, the CP violating asymmetry, $a$,
has a maximum of order -40% to -70% for $\Bar{B}^{0} \to \rho^{0}(\omega)
\pi^{0}$ when the invariant mass of the $\pi^{+}\pi^{-}$ pair is in the
vicinity of the $\omega$ resonance.
  The sensitivity of the asymmetry to $N_{c}^{eff}$ is small in that case.
Moreover, we find that in the range of $N_{c}^{eff}$ which is allowed by the
most recent experimental branching ratios from the BABAR, BELLE and CLEO
Collaborations, the sign of $\sin \delta$ is always positive. Thus, a
measurement of direct CP violation in decays $\Bar{B}^{0} \to \pi^{+}\pi^{-}
\pi^{0}$ would remove the mod$(\pi)$ ambiguity in the determination of the CP
violating phase angle $\alpha$.
