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The magnetism of the 5d metals W and Re is investigated with the method of weak localization. We studied W on the surface of Au and Re on the surface of Ag. For both systems the impurity thicknesses are in the range between 1/100 of a monolayer (single impurities) and one monolayer. For single W and Re impurities the dephasing of weak localization is by two to three orders of magnitude smaller than that of a typical magnetic impurity. For the monolayers it is reduced by a factor of about five. A strong temperature dependence of the dephasing suggests that thermally activated fluctuating moments are present. An extrapolation of this temperature dependence towards T = 0 yields no dephasing and therefore no magnetic behavior at T = 0. Numerical calculations predicted that W and Re impurities possess a magnetic moment of about 3.5 μBon the surface of Ag. These large moments are not found in our experiments.