We consider the full three-body problem describing the motion of three positive masses interacting through their mutual gravitational attraction. We assume that one of the three masses is small and reformulate the system as a perturbation of two uncoupled subsystems: a Kepler problem and a restricted three-body problem, with the small mass serving as the perturbation parameter. We show that the perturbed system exhibits Arnold diffusion, in the sense that there is a transfer of energy -- by an amount independent of the perturbation parameter -- between the Kepler problem and the restricted three-body problem. Our argument is based on the topological method of correctly aligned windows, which is implemented into a computer assisted proof. We show that the approach applies to physically relevant masses of the bodies, choosing a Neptune-Triton-asteroid system as an example. In this case, we obtain explicit estimates for the range of the perturbation parameter and for the time required for the energy transfer. This is based on joint work with Maciej Capinski.