This paper describes the axisymmetric source-sink flow in a rapidly rotating cylinder. Relative fluid motion is induced by the presence of a sink in the bottom corner and a ring source located somewhere in the fluid, at some distance from the solid boundaries. In order to neglect nonlinear effects the volumetric flow rates are assumed to be small, i.e. 0(E1/2), with E the Ekman number of the flow. The transport from the source to the sink is carried by Ekman layers at the end caps, and a Stewartson layer at the sidewall. At the ring source a free Stewartson layer arises, in which the injected fluid is transported towards the Ekman layers. This Stewartson layer consists of layers of thicknesses E 1/4 and E 1/3, which both contribute to the vertical O(E 1/2) transport. The ring source is enveloped by a ring-shaped region of cross-sectional dimensions O(E 1/2 ร El/2), in which the injected fluid is rearranged before erupting into the E 1/3 layer. As E 1/2 << E 1/3, this region appears as an isolated singularity in the E 1/3 layer; in fact it consists of a combination of an upward and a downward directed source, the strengths of which can be determined by transport arguments. The paper presents an analysis of the Ea/3-1ayer structure on the basis of a linear theory; it also describes how the analysis can be extended to the situation in which fluid is injected through an array of sources at different heights.