We examine the effects of curvature on edge-flame propagation speeds as a function of the fuel Lewis number. The diffusion flame supporting the flame edge is assumed to be in Linan's "premixed-flame regime" with oxygen leakage through the flame caused by excessive heat loss to the bounding surface. Adopting an approximate one-dimensional model, we calculate flame-edge propagation velocities in the limit of large activation energy for a one-step gas-phase reaction. Both negative and positive curvatures, corresponding to flame holes and flame disks. respectively, are considered. We predict finite-time singular behavior for both flame-hole closing and flame-disk collapse, meaning that the fronts approach the annihilation with ever-increasing speeds. For Lewis numbers sufficiently large, advancing-edge and retreating-edge solutions are obtained for the same Damkohler number. The results of this study may help to better understand the behavior of spiral flames experimentally observed in swirling flows.