  In e+e- collisions recorded using the CLEO II.V detector we have studied the
Cabibbo suppressed decay of D0 -> pi+pi-pi0 with the initial flavor of the D0
tagged by the decay D*+ -> D0pi+. We use the Dalitz-plot analysis technique to
measure the resonant substructure in this final state and observe \rho\pi and
non-resonant contributions by fitting for their amplitudes and relative phases.
We describe the pipi S-wave with a K-matrix formalism and limit this
contribution to the rate to be < 2.5% @95% confidence level, in contrast to the
large rate observed in D+ -> pi+pi-pi+ decay. Using the amplitudes and phases
from this analysis, we calculate an integrated CP asymmetry of
0.01^{+0.09}_{-0.07} +/- 0.05.
