A new degenerate-kernel approach is developed for the numerical solution of Fredholm integral equations of the second kind, y = f + λKy. An essential feature is that the rank- N approximate kernel is constructed to be exact when it operates in a certain N-dimensional subspace, the subspace being chosen for its suitability for approximating y. The simplest version of the method is equivalent to a single iteration of the method of moments, or Galerkin method, and is very similar to the method of moments in its computational requirements, but nevertheless is not at all similar in its performance. Numerical examples, including one with a logarithmic singularity in the kernel, show that the simplest version gives consistently better results than the method of moments, the errors typically being smaller by one or more factors of ten. A somewhat more elaborate version of the method is found to give results that are at least marginally better again.