he Zhitnitsky and Dine, Fischler and Srednicki (DFSZ) model is a natural extension of the two-Higgs-doublet model containing an additional singlet, endowed with a Peccei-Quinn symmetry, and leading to a physically acceptable axion. In this paper we reexamine this model in the light of some new developments. For generic couplings the model reproduces the minimal Standard Model showing only tiny deviations (extreme decoupling scenario) and all additional degrees of freedom (with the exception of the axion) are very heavy. Recently it has been remarked that the limit where the coupling between the singlet and the two doublets becomes very small is technically natural. Combining this limit with the requirement of exact or approximate custodial symmetry, we may obtain an additional 0+ Higgs at the weak scale, accompanied by relatively light charged and neutral pseudoscalars. The mass spectrum would then resemble that of a generic two-Higgs-doublet model, with naturally adjustable masses in spite of the large scale that the axion introduces. However, the couplings are nongeneric in this model. We use the recent constraints derived from the Higgs-WW coupling together with oblique corrections to constrain the model as much as possible. As an additional result, we work out the nonlinear parametrization of the DFSZ model in the generic case where all scalars except the lightest Higgs and the axion have masses at or beyond the TeV scale.