Abstract <p>For an exterior Neumann problem, a numerical method is proposed based on new quadrature formulas for a single-layer potential, which are based on the analytical evaluation of integrals. The method is tested on the Neumann problem for the Laplace equation outside an ellipsoid, for which explicit solutions are found. It is shown that the numerical solution of the problem produced by the proposed method uniformly approximates the exact solution and ensures a lower error and faster convergence than the numerical solution obtained using standard quadrature formulas based on numerical integration. The properties of numerical solutions depending on the parameters of the ellipsoid are discussed.</p>