We describe an augmented Galerkin technique for the numerical solution of first kind Fredholm equations, which is simple to use and which has the considerable advantage of providing a cheaply computed numerical criterion for the existence of a solution of the equations under study. The method has guaranteed stability, and leads to a standard linear programming problem (when posed in the l1 or lδ norms). It is much faster than the method recommended in a recent review by Lewis (1975); numerical comparisons indicate that it achieves comparable accuracy. The existence criterion also appears effective in practice.