In space missions, scientific data collected by various instruments must be stored onboard before being downlinked to Earth during designated communication windows. For many long range missions, the available bandwidth is shared according to a priority assigned to each memory buffer during such downlink window. The overlapping Memory Dumping Problem (oMDP) consists in finding the priority assignment that minimizes the highest memory peak. This problem has been shown to be weakly NP-hard and has so far only been addressed with heuristic methods. In this paper, we complete the complexity analysis by proving that the problem is strongly NP-hard in the general case, and we propose the first exact method to solve the oMDP. We present a constraint programming approach combining new global constraints and a heuristic branching strategy, and show that our method is competitive with state-of-the-art heuristics while being more generic and able to produce optimality proofs on small instances.