© 2017 American Physical Society. We use x-ray magnetic circular dichroism to evidence the effect of correlations on the local impurity magnetic moment in an archetypal Kondo system, namely, a dilute Cu:Fe alloy. Applying the sum rules on the Fe L2,3 absorption edges, the evolution of the spin and orbital moments across the Kondo temperature are determined separately. The spin moment presents a crossover from a nearly temperature-independent regime below the Kondo temperature to a paramagneticlike regime above. Conversely, the weak orbital moment shows a temperature-independent behavior in the whole temperature range, suggesting different Kondo screening temperature scales for the spin and orbital moments.