We propose a new model for propulsion of the Dictyostelium discoideum grex (pseudoplasmodium). We concentrate upon the mechanics of the problem: how does each participating amoeba contribute motive force, and how do the myriad force contributions produce a coordinated collective effort? Experiments we report here show that when a Dictyostelium discoideum grex's migration is stalled by mechanically arresting the motion of its boundary, the amoebae in it actively circulate in a reverse fountain flow extending the length of the grex. The velocity of individual cells relative to the grex boundary is commensurate with the migration speed of a grex: approximately one grex length per hour. We argue that cell circulation constitutes the propulsive engine of migrating grexes. More precisely, we believe each participating amoeba orients its attempted motion by the same cAMP chemotaxis used during aggregation. The cAMP concentration field within the grex consists of pulses, emitted periodically at the tip, propagating rearward by the same cAMP relay behaviour seen during aggregation. Existing literature documents chemotactic migration within grexes and generally reinforces the preceding description. The principal new contribution of this paper is to resolve the following conceptual difficulty: in a close-packed three-dimensional mass of cells, each amoeba trying to crawl can exert traction only upon its neighbours which, in turn, exert traction on it. In the interior of the grex, with no rigid agar substratum to crawl upon, equal efforts by a cohort of amoebae to crawl in the same direction, each upon similarly crawling neighbours, cancel and produce no net mechanical result. In our model of chemotaxis within packed three-dimensional cell aggregates the cells need no rigid substrate. We hypothesize that gradients of other chemicals must arise naturally within grexes, approximately perpendicular to the average cAMP gradient. If one such chemical acts to modulate the traction amoebae exert individually, then a self-regulating, chemotactically oriented, fountain flow of cells must result. We explain how such a fountain can shape and move the grex, and we speculate on the tendency of the fountain to cause the grex cell mass to climb automatically any stalk that might form inside it. Our work thus strengthens the notion, attractive on evolutionary grounds, that a slight modification of the chemotactic behaviour needed for aggregation can account also for the migratory behaviour of the grex and erection of the fruiting body. In the simplest version of our theory, we assume that regionally differentiated cell types, anterior prestalk and posterior prespore cells, are constantly converting from one to the other. We assume that the grex shapes and moves itself because its constituent cells establish standing chemical gradients, which remain fixed as seen by an observer riding at the grex tip. The cells circulate slowly through these gradients and 'act' in a fashion determined by the local chemical environment, so the same cells continuously interconvert between exhibiting pre-stalk and pre-spore type behaviour. Cell circulation through these standing gradients 'rolls' the grex forward. Of necessity, our formulation is mathematical, phrased in terms of the unfortunately complicated partial differential equations of nonlinear continuum mechanics and spatially heterogeneous reaction-diffusion-convection chemistry. But we give an intuitive discussion, parallel to the essential mathematical one, and show both to predict what we observe experimentally.