A novel extension of a rapid compression machine (RCM), namely a Rapid Compression Expansion Machine (RCEM), is described and its use for studying chemical kinetics is demonstrated. Like conventional RCMs, the RCEM quickly compresses a fuel/air mixture by pushing a piston into a cylinder; the resulting high temperatures and pressures initiate chemical reactions. In addition, the machine can rapidly expand the compressed gas in a controlled way by pulling the piston outwards again. This freezes chemical activity after a pre-defined reaction duration, and therefore allows a convenient probe sampling and ex-situ gas analysis of stable species. The RCEM therefore is a promising instrument for studying chemical kinetics, including also partially reacted fuel/air mixtures. The setup of the RCEM, its experimental characteristics and its use for studying chemical reactions are outlined in detail. To allow comparisons of RCEM results with predictions of chemical reaction mechanisms, a simple numerical model of the RCEM process is described. As a first application, ex-situ measurements of the temporal evolution of species in partially reacted dimethyl ether/methane-mixtures are presented, and the results are compared to predictions of a reaction mechanism.