In this paper we show stability and convergence for a novel Galerkin boundary–element–method approach to the impedance boundary–value problem for the Helmholtz equation in a half–plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree ν) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval [a,b], which only requires the discretization of [a,b], we show theoretically and experimentally that the L2 error in computing the acoustic field on [a,b] is O(logν+3/2|k(b−a)|M−(ν+1), where M is the number of degrees of freedom and k is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems.