Searches for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math>violation and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>wave in the Dalitz-plot analysis of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math>

In ${e}^{+}{e}^{\ensuremath{-}}$ collisions recorded using the CLEO II.V detector we have studied the Cabibbo suppressed decay of ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ with the initial flavor of the ${D}^{0}$ tagged by the decay ${D}^{*+}\ensuremath{\rightarrow}{D}^{0}{\ensuremath{\pi}}^{+}$. We use the Dalitz-plot analysis technique to measure the resonant substructure in this final state and observe $\ensuremath{\rho}\ensuremath{\pi}$ and nonresonant contributions by fitting for their amplitudes and relative phases. We describe the $\ensuremath{\pi}\ensuremath{\pi}$ $S$ wave with a $K$-matrix formalism and limit this contribution to the rate to be $&lt;2.5%@95%$ confidence level, in contrast to the large rate observed in ${D}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ decay. Using the amplitudes and phases from this analysis, we calculate an integrated $CP$ asymmetry of ${0.01}_{\ensuremath{-}0.07}^{+0.09}\ifmmode\pm\else\textpm\fi{}0.05$.