The possibility of having a delocalization transition in the one-dimensional de Moura–Lyra class of models (having a power spectrum ) has been the object of a long-standing discussion in the literature. In this paper, we report numerical evidences that such a transition happens at , where the localization length (measured from the scaling of the conductance) is shown to diverge as . The persistent finite-size scaling of the data is shown to be caused by a very slow convergence of the nearest-neighbor correlator to its infinite-size limit, and controlled by the choice of a proper scaling parameter. Our results for these models are consistent with a localization of eigenstates that is driven by a persistent small-scale noise, which vanishes as . This interpretation is confirmed by analytical perturbative calculations which are built on previous work. Finally, the nature of the delocalization transition is discussed and the conclusions are illustrated by numerical work done in the regime.