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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 accountthe 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 aswell. We find that the CP violating asymmetry is strongly dependent on the CKMmatrix 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 thevicinity 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 themost recent experimental branching ratios from the BABAR, BELLE and CLEOCollaborations, the sign of $\sin \delta$ is always positive. Thus, ameasurement 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 CPviolating phase angle $\alpha$.Comment: 23 pages, 10 figures, 5 tables. Misprints corrected and two figures  update

Direct CP violation in $B \rightarrow \pi^{ + }\pi^{-}\pi$ : determination of $\alpha$ without discrete ambiguity