  We search for CP non-conservation in the decays of tau leptons produced via
$e^+ e^-$ annihilation at $\sqrt{s}\sim$ 10.6 GeV. The method uses correlated
decays of pairs of tau leptons, each decaying to the $\pi \pi^0 \nu_{\tau}$
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 million
produced tau 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 parameterizing the relative contribution of diagrams that
would lead to CP violation to be $-0.046 <\Im(\Lambda) < 0.022$ at 90% C.L.
This result provides a restriction on CP non-conservation in the 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 $\tau\to \pi\pi^0\nu_\tau$ decays. The limit on the imaginary part of
the $\tau$ scalar coupling is $-0.033 < \Im(\Lambda) < 0.089$ at 90% C.L.
