Carbon nanotubes (CNTs) are not intrinsically superconducting but they can carry a supercurrent when connected to superconducting electrodes 1-4. This supercurrent is mainly transmitted by discrete entangled electron-hole states confined to the nanotube, called Andreev bound states (ABS). These states are a key concept in mesoscopic superconductivity as they provide a universal description of Josephson-like effects in quantum-coherent nanostructures (for example molecules, nanowires, magnetic or normal metallic layers) connected to superconducting leads 5. We report here the first tunnelling spectroscopy of individually resolved ABS, in a nanotubesuperconductor device. Analysing the evolution of the ABS spectrum with a gate voltage, we show that the ABS arise from the discrete electronic levels of the molecule and that they reveal detailed information about the energies of these levels, their relative spin orientation and the coupling to the leads. Such measurements hence constitute a powerful new spectroscopic technique capable of elucidating the electronic structure of CNT-based devices, including those with well-coupled leads. This is relevant for conventional applications (for example, superconducting or normal transistors, superconducting quantum interference devices 3 (SQUIDs)) and quantum information processing (for example, entangled electron pair generation 6,7 , ABS-based qubits 8). Finally, our device is a new type of d.c.measurable SQUID. First conceived of four decades ago 9 , ABS are electronic analogues of the resonant states in a Fabry-Pérot resonator. The cavity is here a nanostructure and its interfaces with superconducting leads play the role of the mirrors. Furthermore, these 'mirrors' behave similarly to optical phase-conjugate mirrors: because of the superconducting pairing, electrons in the nanostructure with energies below the superconducting gap are reflected as their time-reversed particle-a process known as Andreev reflection. As a result, the resonant standing waves-the ABS-are entangled pairs of timereversed electronic states, which have opposite spins (Fig. 1a); they form a set of discrete levels within the superconducting gap (Fig. 1b) and have fermionic character. Changing the superconducting phase difference ϕ between the leads is analogous to moving the mirrors and changes the energies E n (ϕ) of the ABS. In response, a populated ABS carries a supercurrent (2e/h)(∂E n (ϕ)/∂ϕ) through the device, whereas states in the continuous spectrum (outside the superconducting gap) have negligible or minor contributions in most common cases 5. Therefore, the finite set of ABS generically determines Josephson-like effects in such systems. As such, ABS