The efficient transfer of energy across different quantum systems - an essential ingredient to quantum-information-processing protocols - requires that the coupling between these systems is stronger than the decay rate of either one. Here we report experiments in which we have reached this so-called strong-coupling regime in a hybrid system consisting of a gate-defined double quantum dot and a superconducting resonator. These results underline the potential of circuit-QED experiments with semiconductor nano-structures both for quantum information processing and for fundamental studies of light–matter interactions.