Summary: We construct a rigorous renormalization scheme for two-degree-of-freedom analytic Hamiltonians associated with Diophantine frequency vectors. We prove the existence of an attracting, integrable limit set of the renormalization. As an application of this renormalization scheme, we give a proof of a KAM theorem. We construct analytic invariant tori with Diophantine frequency vectors for near-integrable Hamiltonians in the domain of attraction of this limit set.