The Josephson effect describes the flow of supercurrent in a weak link-such as a tunnel junction, nanowire or molecule-between two superconductors(1). It is the basis for a variety of circuits and devices, with applications ranging from medicine(2) to quantum information(3). Experiments using Josephson circuits that behave like artificial atoms(4) are now revolutionizing the way we probe and exploit the laws of quantum physics(5,6). Microscopically, the supercurrent is carried by Andreev pair states, which are localized at the weak link. These states come in doublets and have energies inside the superconducting gap(7-10). Existing Josephson circuits are based on properties of just the ground state of each doublet, and so far the excited states have not been directly detected. Here we establish their existence through spectroscopic measurements of superconducting atomic contacts. The spectra, which depend on the atomic configuration and on the phase difference between the superconductors, are in complete agreement with theory. Andreev doublets could be exploited to encode information in novel types of superconducting qubits(11-13).