A new boundary integral equation (BIE) method for the numerical solution of the exterior Neumann problem for the Laplace equation in planar domains with corners is proposed. Using the single layer representation of the potential, the differential problem is reformulated in terms of a BIE. In order to address the more challenging problems of proving the stability and convergence of the numerical method, arising in this context, a “modified'” Nyström method, based on a Gaussian quadrature formula, is proposed in order to approximate the solution of the BIE and, consequently, the single layer potential. Moreover, suitable regularization techniques are introduced that improve the smoothness of the solution and, consequently, the convergence rate of the approximation.