Galerkin's method is to be used to solve a Fredholm integral equation of the first kind in which the kernel has a logarithmic singularity, and the path of integration is either a polygon or an open arc. The singularities of the solution, due to corners of the boundary or end points of the arc, will adversely affect the rate of convergence of the approximate solution to the true solution. It is shown that when the space of approximating functions is generated by piecewise constants then, in some cases, the expected \(O(h^ 3)\) order of convergence can be restored by a suitable grading of the mesh near the corners of the polygon or the endpoints of an open arc. Three examples, when the path of integration is the unit interval, the unit square and the re-entrant L-shaped region, are examined in detail.