We present a conservative method for the three-dimensional coupling between an inviscid compressible flow and a moving rigid solid. We consider an inviscid Euler fluid in conservativeform discretized using a high-order monotonicity-preserving Finite Volume method with a directional operator splitting.An Immersed Boundary technique is employed through the modification of the Finite Volume fluxesin the vicinity of the solid.The method yields exact conservation of mass, momentum and energy of the system, andalso exhibits important consistency properties, such as conservation of uniform movement of both fluid and solid as well as the absence of numerical roughness on a straight boundary. The coupling scheme evaluates the fluxes on the fluid side and theforces and torques on the solid side only once every time step, ensuring the computational efficiency of the coupling.We present numerical results assessing the robustness of the method in the case of rigid solids with large displacements.