We have performed in situ reflectivity measurements using synchrotron radiation of Ag films deposited on Ge(111) over the thickness range of 3–12 atomic layers. The films deposited at a substrate temperature of 110 K are not well ordered, but become well ordered upon annealing, as evidenced by substantial changes in the x-ray reflectivity data. The thickness distribution for each annealed film, deduced from a fit to the reflectivity data, is remarkably narrow, with just two or three adjacent discrete thicknesses present, despite the large lattice mismatch between Ag and Ge. In some cases, the film thickness is nearly atomically uniform. The results are discussed in connection with recent models and theories of electronic effects on the growth of ultrathin metal films.