The geology of northern and central Africa is characterized by broad plateaux, narrower swells and volcanism occurring from ~45Myr ago to the present. The greatest magma volumes occur on the >1,000-km-wide Ethiopian and east African plateaux, which are transected by the Red Sea, Gulf of Aden and east African rift systems, active since the late Oligocene epoch. Evidence for one or more mantle plumes having impinged beneath the plateaux comes from the dynamic compensation inferred from gravity studies, the generally small degrees of extension observed and the geochemistry of voluminous eruptive products. Here we present a model of a single large plume impinging beneath the Ethiopian plateau that takes into account lateral flow and ponding of plume material in pre-existing zones of lithospheric thinning. We show that this single plume can explain the distribution and timing of magmatism and uplift throughout east Africa. The thin lithosphere beneath the Mesozoic-Palaeogene rifts and passive margins of Africa and Arabia guides the lateral flow of plume material west to the Cameroon volcanic line and south to the Comoros Islands. Our results demonstrate the strong control that the lithosphere exerts on the spatial distribution of plume-related melting and magmatism.