A stable and high--order method for solving the Helmholtz equation on a two--dimensional domain exterior to a bounded obstacle is developed in this paper. The method is based on a boundary perturbation technique (``Transformed Field Expansions'') coupled with a well--conditioned high--order spectral--Galerkin solver. The method is further enhanced with numerical analytic continuation implemented via Pade approximation. Ample numerical results are presented to show the accuracy, stability, and versatility of the proposed method.