The growth of reactive adatom-substrate overlayers has been examined during the deposition of Ag and Cu on Bi(0001) using primarily Auger electron spectroscopy (AES) and low-energy electron diffraction (LEED). When viewed by the commonly used signal-versus-time plot, the Auger data exhibit a featureless behavior with a temperature dependence that is suggestive of cluster or compound formation. Surprisingly, when the same data are portrayed on a substrate-versus-adatom plot of the signals, the results are quite unlike those previously reported in the literature, in that the curvature of the data is the opposite of that typically found as the adatom signal increases. Heavy adatom deposition is needed to totally obscure the substrate signal. In the absence of factors that are known to affect the Auger yield in such overlayer systems, these data are found to be consistent with a growth process in which a layer of Bi substrate atoms is continuously segregated at the vacuum-solid interface. This interpretation is consistent with predictions from surface free-energy arguments, in that the Bi has a much lower energy than Ag or Cu and, thus, has a strong tendency to stay on top of the growing film.