In this work, we treat theoretically the conjugate film-condensation process on a vertical fin immersed in a saturated vapor, taking into account that the viscosity of the condensed film is a well-defined function of the temperature. In order to predict the thickness of the film, the momentum and energy balance equations for the condensate and the energy equation for the vertical fin are reduced to a nonlinear system of two ordinary differential equations. These governing equations contain four nondimensional parameters, the Jakob number, Ja, a conjugate heat transfer parameter, alpha, the aspect ratio of the fin, epsilon, and (beta) over bar, that take into account the effect of the variable viscosity. Using the limit Ja << 1 and the boundary layer approximation for the film-condensation process, the nondimensional heat transfer and the overall mass flow rates of the condensed fluid have been obtained as functions of the involved nondimensional parameters.