We propose an algorithm for the numerical solution of the Lur'e equations in the bounded real and positive real lemma for stable systems. The algorithm provides approximate solutions in low-rank factored form. We prove that the sequence of approximate solutions is monotonically increasing with respect to definiteness. If the shift parameters are chosen appropriately, the sequence is proven to be convergent to the minimal solution of the Lur'e equations. The algorithm is based on the ideas of the recently developed ADI iteration for algebraic Riccati equations. In particular, the matrices obtained in our iteration express the optimal cost in a certain projected optimal control problem.