A model of one dimensional grain growth in a bicrystal film, driven by surface anisotropy, is presented. The equation of surface diffusion (Mullins Equation) and the curvature driven boundary equation of motion are simultaneously solved under a set of matching and boundary conditions that includes the mechanical equilibrium at the discontinuous points (surface corners due to anisotropy and grain boundary groove roots), the conservation of surface flux and the continuity of chemical potential at the groove root. An analytical solution is obtained for the steady state motion. This solution is investigated as a function of the degree of anisotropy. It is found that a steady state motion is possible for a limited range of anisotropies. Below a critical degree of anisotropy the film breaks up. It is shown that the steady state velocity decreases with the degree of anisotropy. Numerical analysis of the time dependent problem demonstrates that the steady state solution is stable. It is shown that anisotropy stabilizes the process of grain growth against film breakup. ©1997 American Institute of Physics.