Consider an analytic Hamiltonian system near its analytic invariant torus T-0 carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at T-0 is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation-not just a formal power series-bringing the Hamiltonian into its Birkhoff normal form.