The propagation of small-amplitude nonlinear Alfven waves on a single magnetic interface in a cold plasma is considered. The anisotropic ion viscosity is taken into account. The governing equation is derived with the use of the reductive perturbation method. The evolution of an initially sinusoidal wave with the distance from the source is studied numerically. Calculations show the wave shape is very steep at some distance from the source if the viscosity is very small. This wave steepening leads to strong acceleration of wave damping. The dependencies of wave energy flux on the distance from the wave source is calculated for different values of the Reynolds number which determines the ratio of the nonlinearity effect to the viscosity one. This formula shows that in the case of small viscosity the wave-damping distance predicted by the nonlinear theory can be one order of magnitude or more smaller than one predicted by the linear theory. 15 refs.