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The magnetism of quench-condensed Ru and Rh impurities and metal films on Ag(100) and Pt(997) has been studied using x-ray magnetic circular dichroism. In the coverage range between 0.22 and 2.0 ML, no dichroic signal was detected at the M3,2 absorption edges of Ru on Ag(100) at a temperature of 5 K in the presence of an applied magnetic field. The same was found for coverages between 0.12 and 0.5 ML of Rh on Ag(100) and Pt(997). It is concluded that the magnetic moments of single impurities, small clusters of various shape, and monolayers of the 4d metals are below the detection limit of 0.04 μB per atom. These results provide an unambiguous determination of the local magnetic moment of Ru and Rh deposited on nonmagnetic transition-metal surfaces, which are in contrast with theoretical predictions.