This paper is the second part of a work on Laplace and Helmholtz integral equations in 2 space dimensions on open curves. A new Galerkin method in weighted L 2 spaces together with new preconditioners for the weighted layer potentials are studied. This second part provides the theoretical analysis needed to establish the results announced in the first part. The main novelty is the introduction of a pseudo-differential calculus on open curves that allows to build parametrices for the weighted layer potentials. Contrarily to more classical approaches where the Mellin transform is used, this new approach is well-suited to the specific singularities that appear in the problem.