The natural frequencies and damping rates of surface waves in a circular cylinder with pinned-end boundary conditions are calculated in terms of the gravitational Reynolds and Bond numbers, C−1 and B, and the slenderness of the cylinder Λ, in the limit C→0. We consider higher-order approximations that include the effect of viscous dissipation in the Stokes boundary layers and the bulk. A comparison with clean-surface experiments by Henderson & Miles (1994) shows a satisfactory agreement except for the first axisymmetric mode, which exhibits a 26% discrepancy. The much larger dramatic discrepancy of former theoretical predictions is hereby improved and explained.