An approximation method for high-energy potential scattering is developed that expresses the scattered amplitude in terms of a quadrature, similar to the Born approximation but superior to it in accuracy. It is valid when the potential is slowly varying compared to a wavelength, $\frac{|V|}{{E}^{\ensuremath{'}}}$ is small compared to unity, $\ensuremath{\theta}$ is either small or large compared to ${(\mathrm{kR})}^{\ensuremath{-}\frac{1}{2}}$, and $\frac{|V|R}{\ensuremath{\hbar}v}$ is unrestricted in magnitude, where ${E}^{\ensuremath{'}}$, $\ensuremath{\theta}$, $k$, and $v$ are the kinetic energy, scattering angle, wave number, and speed of the scattered particle, and $V$ and $R$ are rough measures of the strength and range of the scattering potential, which may be complex. For comparison, the Born approximation requires that $\frac{|V|R}{\ensuremath{\hbar}v}$ be small compared to unity. The procedure consists in summing the infinite Born series after approximating each term by the method of stationary phase. Both the Schr\"odinger and Dirac equations are treated, and it is expected that the method can be extended to the scattering theory of other wave equations. The relation of the present work to previous work of others is discussed, and the limitations of WKB or eikonal-type approximations are explored. The method is expected to be especially useful for calculating the scattering of fast electrons, neutrons, and protons from nonspherical nuclei.