Evidence for a destructive interference effect between the ${\rho }^0$ and $\omega \to {\pi }^{+}{\pi }^{-}$ amplitudes has been observed in the channel ${\pi }^{+}p\to {\pi }^{+}{\pi }^{-}{\Delta }^{++}$ at 3.7-4.0 GeV/c. On the assumption of complete coherence between the ${\rho }^0$ and $\omega \to {\pi }^{+}{\pi }^{-}$ amplitude in a subsample, discussed in the text, a branching ratio $\frac{(\omega \to {\pi }^{+}{\pi }^{-})}{(\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0)}=1.5\%$ is obtained with a 95% confidence-level lower limit of 0.25%. The data do not lead to a significant upper limit. In terms of $\delta$, the off-diagonal element of the ${\rho }^0\omega$ mass-mixing matrix, these values are equivalent to $\delta =2.7\mathrm{}$ MeV and a lower limit of 1.1 MeV. Under conditions stated in the text, a value $\beta =1.5\mathrm{}\pm 0.3$ rad is obtained for the production phase of ${\rho }^0$ relative to $\omega$ in the channel studied.

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