The divergence of new gene functions is described by various scenarios that involve gene duplication, albeit, at fundamentally different stages. We performed experimental measurements and developed a subsequent model, aimed at predicting the rate of divergence under different scenarios. We used gene libraries of TEM-1 beta-lactamase that were drifted under purifying selection toward the original penicillinase activity or under no selection at all. The frequency of genes conferring a new function (degradation of a cephalosporin antibiotic) was measured at various stages of the drift, and a model that accounts for the differences in the observed adaptation dynamics of the drifting TEM-1 populations was derived. The results indicate that rapid nonfunctionalization in the population relieved from selection (Ohno's model) afforded only a narrow window of adaptation to cefotaxime (neofunctionalization). The trade-off between TEM-1's original function and the new evolving function also disfavored the "gene sharing" model. The rate of adaptation was maximal when selection for the original function was partially relieved to enable the accumulation of potentially adaptive mutations, while still purging a large fraction of otherwise deleterious mutations. Altogether, scenarios of subfunctionalization seem more feasible, whereby sustaining the original function by two copies facilitates the accumulation potentially adaptive mutations while purging nonfunctionalization mutations.