The feasibility of simultaneous removal and concentration of H2S from natural gas by pressure swing adsorption (PSA) is established. The predried feed contains 1000 ppm H2S and 5% CO2, The purified adsorption product contains below 1 ppm H2S at methane product recoveries above 95%. At the same time a desorption product with 2% H2S is obtained which can be readily converted into sulfur by known processes. In the PSA system, there is a region of feed velocities where two stable cyclic steady states and an unstable steady state exist. Under the same PSA conditions considered, the product impurities in the two stable steady states are approximately 100 and 1 ppm H2S, depending on the initial bed conditions. Upon examining the differences between the H2S concentration profiles in the bed at the two steady states, the lower steady state is described by a single wave kept short from breaking through the bed, while the upper steady state is described by a combined wave with transfer zones at each end of the bed, connected by a "middle" constant zone.