The main issue of the paper has consisted in fixing the link between the vacuum-energy cancellation and the present Universe dark energy. It is assumed that the actual acceleration is due to the relic of the original huge vacuum-energy, after its mean value has been compensated for by the intrinsic cosmological constant contained in the exponential Lagrangian. The free parameters of such a Langrangian density have been fixed as functions of the cosmological constant, and, in the de Sitter regime, the ratio between the vacuum-energy density and the geometrical contribution has been illustrated to acquire a negative sign. In particular, the cutoff introduced in the vacuum-energy density has been linked with the modified commutation relation following from a generalized uncertainity principle, and has been fixed at Planck scales.