Gas-filled stopping cells are used at accelerator laboratories for the thermalization of high-energy radioactive ion beams. Common challenges of many stopping cells are a high molecular background of extracted ions and limitations of extraction efficiency due to space-charge effects. At the FRS Ion Catcher at GSI, a new technique for removal of ionized molecules prior to their extraction out of the stopping cell has been developed. This technique utilizes the RF carpet for the separation of atomic ions from molecular contaminant ions through their difference in ion mobility. Results from the successful implementation and test during an experiment with a 600 MeV/u 124 Xe primary beam are presented. Suppression of molecular contaminants by three orders of magnitude has been demonstrated. Essentially background-free measurement conditions with less than 1% of background events within a mass-to-charge range of 25 u/e have been achieved. The technique can also be used to reduce the space-charge effects at the extraction nozzle and in the downstream beamline, thus ensuring high efficiency of ion transport and highly-accurate measurements under space-charge-free conditions.