A paradigm of cytokinesis in animal cells is that the actomyosin contractile ring provides the primary force to divide the cell. In the fission yeast Schizosaccharomyces pombe, cytokinesis also involves a conserved cytokinetic ring, which has been generally assumed to provide the force for cleavage (see also [5]). However, in contrast to animal cells, cytokinesis in yeast cells also requires the assembly of a cell wall septum, which grows centripetally inward as the ring closes. Fission yeast, like other walled cells, also possess high (MPa) turgor pressure. Here, we show that turgor pressure is an important factor in the mechanics of cytokinesis. Decreasing effective turgor pressure leads to an increase in cleavage rate, suggesting that the inward force generated by the division apparatus opposes turgor pressure. The contractile ring, which is predicted to provide only a tiny fraction of the mechanical stress required to overcome turgor, is largely dispensable for ingression; once septation has started, cleavage can continue in the absence of the contractile ring. Scaling arguments and modeling suggest that the large forces for cytokinesis are not produced by the contractile ring but are driven by the assembly of cell wall polymers in the growing septum.