We present a reliable double-folding (DF) model for He4-nucleus scattering, using the Melbourne g-matrix nucleon-nucleon interaction that explains nucleon-nucleus scattering with no adjustable parameter. In the DF model, only the target density is taken as the local density in the Melbourne g matrix. For He4 elastic scattering from Ni58 and Pb208 targets in a wide range of incident energies from 20 to 200 MeV/nucleon, the DF model with the target-density approximation (TDA) yields much better agreement with the experimental data than the usual DF model with the frozen-density approximation in which the sum of projectile and target densities is taken as the local density. We also discuss the relation between the DF model with the TDA and the conventional folding model in which the nucleon-nucleus potential is folded with the He4 density.