Low-Rossby-number flow past a circular cylinder in a rapidly rotating frame is studied where N is equal to E in terms of the Ekman number E and Rossby number Ro. For this parameter range the boundary layer contains a singularity at the rear stagnation point. The asymptotic structure of this singularity is shown to consist of three distinct asymptotic regions, one of which is viscous while the others are inviscid. New accurate numerical solutions of the boundary-layer equation confirm this singularity structure. The use of Von Mises coordinates both simplifies the analysis, and enables numerical solutions to be found closer to the critical value N = 1, beneath which the flow separates upstream of the rear stagnation point.