An apparatus and method for compressing a cryogenic media is disclosed. The apparatus includes a compressor compartment surrounded by a cylinder wall, a compressor piston being moved linearly in the compartment, an intake valve, and a compression valve, both valves being arranged in the area of the lower end position of the compressor piston. The compression valve is arranged laterally on the cylinder wall in the area of the lower end position of the compressor piston, and the head of the compressor piston has a conical shape.