Numerous orbits in the solar system and in astrodynamics exhibit very peculiar motions. Their common feature is that they involve two moons or satellites orbiting around a much heavier central attractor with almost equal semi-major axes, this is called a co-orbital motion. Despite analytical theories and numerical investigations showing their long-term stability, very few rigorous results in this direction have been obtained so far. In the considered case, there is a singularity which prevents the application of standard perturbation theory for the three-body problem. Adapting classical ideas of Arnold (1963) in this context and, more specifically, by applying KAM theory to the planar three-body problem, we provide a rigorous proof of the existence of a large measure set of invariant tori supporting quasi-periodic co-orbital motions, hence stable over infinite times.