We study the growth of first-layer islands making use of a mean-field model to describe the diffusion of free adatoms on the uncovered substrate. Model parameters are determined from recent experimental results for the growth of metals. The growth law is found using the quasi-steady-state approximation and we show that this result is consistent, at short times where this approximation may be questioned, with the exact asymptotic result. \textcopyright{} 1996 The American Physical Society.