A new method is introduced for the solution of problems of scattering by rough surfaces in the high-frequency regime. It is shown that high-order summations of expansions in inverse powers of the wave number can be used within an integral equation framework to produce highly accurate results for surfaces and wavelengths of interest in applications. Our algorithm is based on systematic use and manipulation of certain Taylor-Fourier series representations and explicit asymptotic expansions of oscillatory integrals. Results with machine precision accuracy are presented which were obtained from computations involving expansions of order as high as 20.