Accurate numerical evaluation of boundary integrals is fundamental to producing useful results with the boundary element method. This paper introduces a generalisation of a recently introduced combined method (subtraction of singularity followed by a non-linear transformation), which takes into account the effect of the basis functions. The new method is applied to solve weakly singular integrals which arise in the solution of the two-dimensional Laplace equation. The new method was found, in the cases considered, to be numerically superior to both the combined method and any of the non-linear transformation methods.