Non-leptonic hyperon weak decays are investigated in the SU(3) Skyrme model. We use a collective coordinate scheme, following the approach in which the symmetry breaking terms in the strong effective action are diagonalized exactly. To describe the weak interactions we use an octet dominated weak effective lagrangian that leads to a good description of the known 2π and 3π kaon decays. We show that the observed S-wave decays are reasonably well reproduced in the model. On the other hand, our calculated P-wave amplitudes do not agree with the empirical ones even though both pole and contact contributions to these amplitudes are properly taken into account. Finally, an estimate of the non-octet contributions to the decay amplitudes is presented.