The objective of this work is to present a conservative coupling method between an inviscid compressible fluid and a deformable structure. The coupling hinges on a Conservative Immersed Boundary method in combination with a Finite Volume method for the fluid and a Discrete Element method for the deformable structure. A time semi-implicit approach is used for the computation of the energy and momentum transfer between the solid and the fluid. The coupling method yields exact conservation of mass, momentum, and energy of the system, and also exhibits consistency properties, such as conservation of uniform movement of both fluid and solid, absence of numerical roughness on a straight boundary, and preservation of a constant fluid state around a wall having tangential deformation velocity. The performance of the method is assessed on test cases involving two and three-dimensional deformable solids with large displacements interacting with shocked fluids.