We examine the stability of a thin film of viscous fluid inside a cylinder with horizontal axis, rotating about this axis. Depending on the parameters involved, the dynamics of the film can be described by several asymptotic equations, one of which was examined by Benilov, O'Brien, and Sazonov (J. Fluid Mech. 2003 497, 201-224). It turned out that the linear stability problem for this equation admits infinitely many harmonic eigenmodes, which are all neutrally stable. Despite that, the film is unstable with respect to 'exploding' (non-harmonic) disturbances, which grow infinitely in finite time. The present paper examines the effect of surface tension on the stability of the film. Given the generally stabilizing nature of surface tension, it comes as no surprise that it eliminates the exploding solutions and makes most eigenmodes asymptotically (not just neutrally) stable. For a certain parameter range, however, some of the eigenmodes, paradoxically, become unstable.