A solution of the plane problem of the theory of elasticity for a film–substrate composite is solved by a perturbation method for a substrate with a rough surface. An algorithm for calculating any approximation, which ultimately leads to the solution of the same Fredholm equation of the second kind, is given. Formulae for calculating the right-hand side of this equation, which depends on all the preceding approximations, are derived. An exact solution of the integral equation in the form of Fourier series, whose coefficients are expressed in quadratures, is given in the case of a substrate with a periodically curved surface. The stresses on the flat surface of the film and on the interfacial surface are found in a first approximation as functions of the form of bending of the surface, the mean thickness of the film and the ratio of Young's moduli of the film and the substrate. It is shown, in particular, that the greatest stress concentration on the film surface occurs on a protrusion of the softer substrate.