A rapidly convergent boundary‐element method that has a high‐accuracy capability is developed for the solution of the exterior Neumann problem for the Helmholtz equation. The approach makes use of the boundary‐operator combination idea of Burton and Miller [A. J. Burton and G. F. Miller, Proc. R. Soc. London Ser. A323, 201–210 (1971)] to avoid the classical irregular frequency difficulties together withp‐version boundary elements to obtain a high rate of convergence. The entire algorithm is designed to be fully hierarchic to minimize the cost of multiple solutions, which are necessary foraposterioriassessment of accuracy. A new hierarchic singular‐kernel quadrature rule is developed for this purpose. Numerical examples demonstrate the accuracy and convergence rate of the method.