Summary: A new product integration scheme is proposed to solve Hammerstein equations which are weakly singular. The standard way of implementing the product integration method to a nonlinear equation is to transform the functional equation to an nonlinear finite dimensional algebraic system by the product integration scheme and then linearize the system to solve it. In this paper, we propose to treat the nonlinearity first. We construct a Newton sequence in the infinite dimensional functional space. Then we approximate the Newton iterates by the product integration method. We prove that the iterates, issued from our method, tend to the exact solution of the nonlinear Hammerstein equation when the number of Newton iterations tends to infinity, whatever the mesh size can be. This is not the case when the discretization is done first: in this case, the accuracy of the approximation is limited by the mesh size.