A singular Fredholm equation of the second kind is solved numerically by a Fourier series analysis in which the singularity is removed naturally, and by a Gaussian quadrature procedure in which the singularity was eliminated by an approximation using the law of the mean. In addition, an analytic solution of an associated Fredholm equation of the first kind was used for comparison with the numerical results. Some properties of these numerical solutions are indicated, and a brief discussion of the errors is given.