We search for CP nonconservation in the decays of τ leptons produced via e+e- annihilation at √s ∼ 10.6 GeV. The method uses correlated decays of pairs of τ leptons, each decaying to the ππ0ντ final state. The search is done within the framework of a model with a scalar boson exchange. In an analysis of a data sample corresponding to 12.2×106 produced τ pairs collected with the CLEO detector, we find no evidence of violation of CP symmetry. We obtain a limit on the imaginary part of the coupling constant parametrizing the relative contribution of diagrams that would lead to CP violation to be -0.046<Im(Λ)<0.022 at 90% C.L. This result provides a restriction on CP nonconservation in tau lepton decays. As a cross-check, we study the decay angular distribution and perform a model-independent search for a CP violation effect of a scalar exchange in single τ→ππ0ντ decays. The limit on the imaginary part of the τ scalar coupling is -0.033 <Im(Λ)<0.089 at 90% CL.