The culmination of the morphogenesis of the cellular slime mould Dictyostelium discoideum involves complex cell movements which transform a mound of cells into a globule of spores on a slender stalk. We show that cyclic AMP signalling and differential adhesion, combined with cell differentiation and slime production, are sufficient to produce the morphogenetic cell movements which lead to culmination. We have simulated the process of culmination using a hybrid cellular automata/partial differential equation model. With our model we have been able to reproduce the main features that occur during culmination, namely the straight downward elongation of the stalk, its anchoring to the substratum and the formation of the long thin stalk topped by the spore head. We conclude that the cyclic AMP signalling system is responsible for the elongation and anchoring of the stalk, but in a roundabout way: pressure waves that are induced by the chemotaxis towards cyclic AMP squeeze the stalk through the cell mass. This mechanism forces the stalk to elongate precisely in the direction opposite to that of the chemotactically moving cells. The process turns out to be 'guided' by inactive 'pathfinder' cells, which form the tip of the stalk. We show that the entire development is enacted by means of the aforementioned building blocks. This means that no global gradients or different modes of chemotaxis are needed to complete the culmination.