This report describes the progress made on the design of the cryogenic cooling system for the liquid absorber for the international Muon Ionization Cooling Experiment (MICE). The absorber consists of a 20.7-liter vessel that contains liquid hydrogen (1.48 kg at 20.3 K) or liquid helium (2.59 kg at 4.2 K). The liquid cryogen vessel is located within the warm bore of the focusing magnet for the MICE. The purpose of the magnet is to provide a low beam beta region within the absorber. For safety reasons, the vacuum vessel for the hydrogen absorber is separated from the vacuum vessel for the superconducting magnet and the vacuum that surrounds the RF cavities or the detector. The absorber thin windows separate the liquid in the absorber from the absorber vacuum. The absorber vacuum vessel also has thin windows that separate the absorber vacuum space from adjacent vacuum spaces. Because the muon beam in MICE is of low intensity, there is no beam heating in the absorber. The absorber can use a single 4 K cooler to cool either liquid helium or liquid hydrogen within the absorber.