The authors have built a microwave Fabry-Perot resonator made of diamond-machined copper mirrors coated with superconducting niobium. Its damping time (T-c=130 ms at 51 GHz and 0.8 K) corresponds to a finesse of 4.6x10(9), the highest ever reached for a Fabry-Perot in any frequency range. This result opens many perspectives for quantum information processing, decoherence and nonlocality studies.