  In e^+e^- collisions using the CLEO detector we have studied the decay of the
D^0 to the final state K_S^0 pi^+ pi^- with the initial flavor of the D^0
tagged in charged D^* decay. We use the Dalitz technique to measure the
resonant substructure in this final state and clearly observe ten different
contributions by fitting for their amplitudes and relative phases. We observe a
K^{* +} pi^- component which arises from doubly Cabibbo suppressed decays or
D^0 - {D^0}-bar mixing.
