Weakly nonlinear stability theory is used to study the stability characteristics of condensate film flowing down the outer surface of a vertical cylinder. The surface tension and the mass transfer due to phase change are taken into account at the liquid-vapor interface. A method of perturbation is applied for the solution and the results show that supercritical stability in the linearly unstable region and subcritical instability in the linearly stable region exist. The lateral curvature of the cylinder has a destabilizing effect on the flow stability. The curvature of the cylinder will intensify the instability of the film flow in comparison with that of the planar flow. A possible application of the present results to some aspects of the qualitative design of a coating process is given.