We have built a microwave Fabry-Perot resonator made of diamond-machined copper mirrors coated with superconducting niobium. Its damping time (Tc = 130 ms at 51 GHz and 0.8 K) corresponds to a finesse of 4.6 e9, the highest ever reached for a Fabry-Perot in any frequency range. We have tested this resonator by sending across it two circular Rydberg atoms, the first emitting a photon and the second absorbing it after a delay of 1/10 s. This long storage time photon box opens novel perspectives for quantum information. It can be used to perform sequences of hundreds of gate operations on individual atomic qubits. A set-up with one or two photon boxes can store mesoscopic fields made of hundreds of photons for decoherence and non-locality studies.