We calculate the nonleptonic weak decays of hyperons in the Skyrmion model of baryons. The weak-interaction chiral Lagrangian is that known from kaon decay, providing a zero-parameter description of the decay amplitudes. In the S waves the PCAC (partially conserved axial-vector current) relation is verified, but the magnitude emerges as a factor of 3 too small, probably indicating a slow convergence of the chiral expansion. In the P waves, new and important terms are found which are not included in standard analyses. These modify the PCAC treatment of hyperon decay and can provide the resolution of the long-standing S-wave/P-wave puzzle.