Methods described to date for the solution of linear Fredholm integral equations have a computing time requirement of O(N3), where N is the number of expansion functions or discretization points used. We describe here a Tchebychev expansion method, based on the FFT, which reduces this time to O(N2 ln N), and report some comparative timings obtained with it. We give also both a priori and a posteriori error estimates which are cheap to compute, and which appear more reliable than those used previously.