The extent to which an emerging new function trades off with the original function is a key characteristic of the dynamics of enzyme evolution. Various cases of laboratory evolution have unveiled a characteristic trend; a large increase in a new, promiscuous activity is often accompanied by only a mild reduction of the native, original activity. A model that associates weak trade-offs with "evolvability" was put forward, which proposed that enzymes possess mutational robustness in the native activity and plasticity in promiscuous activities. This would enable the acquisition of a new function without compromising the original one, reducing the benefit of early gene duplication and therefore the selection pressure thereon. Yet, to date, no experimental study has examined this hypothesis directly. Here, we investigate the causes of weak trade-offs by systematically characterizing adaptive mutations that occurred in two cases of evolutionary transitions in enzyme function: (1) from phosphotriesterase to arylesterase, and (2) from atrazine chlorohydrolase to melamine deaminase. Mutational analyses in various genetic backgrounds revealed that, in contrast to the prevailing model, the native activity is less robust to mutations than the promiscuous activity. For example, in phosphotriesterase, the deleterious effect of individual mutations on the native phosphotriesterase activity is much larger than their positive effect on the promiscuous arylesterase activity. Our observations suggest a revision of the established model: weak trade-offs are not caused by an intrinsic robustness of the native activity and plasticity of the promiscuous activity. We propose that upon strong adaptive pressure for the new activity without selection against the original one, selected mutations will lead to the largest possible increases in the new function, but whether and to what extent they decrease the old function is irrelevant, creating a bias towards initially weak trade-offs and the emergence of generalist enzymes.