In this paper we investigate the effects of element diffusion on the structure and evolution of low-mass helium white dwarfs. Attention is focused mainly on the occurrence of hydrogen-shell flashes induced by diffusion processes during cooling phases. Physically sound initial models with stellar masses of 0.406, 0.360, 0.327, 0.292, 0.242, 0.196, 0.169 and 0.161 M⊙ are constructed by applying mass-loss rates at different stages of the red giant branch evolution of a solar model up to the moment the model begins to evolve to the blue part of the HR diagram. The multicomponent flow equations describing gravitational settling, and chemical and thermal diffusion are solved and the diffusion calculations are coupled to an evolutionary code. In addition, the same sequences are computed but neglecting diffusion. Results without diffusion are similar to recent results of Driebe, Schönberner, Blöcker and Herwig. We find that element diffusion strongly affects the structure and cooling history of helium white dwarfs. In particular, diffusion induces the occurrence of hydrogen-shell flashes in models with masses ranging from 0.18 to 0.41 M⊙, which is in sharp contrast with the situation when diffusion is neglected. In connection with further evolution, these diffusion-induced flashes lead to much thinner hydrogen envelopes, preventing stable nuclear burning from being a sizeable energy source at advanced stages of evolution. This implies much shorter cooling ages than in the case when diffusion is neglected. These new evolutionary models are discussed in light of recent observational data on some millisecond pulsar systems with white dwarf companions. In this context, we find that discrepancies between spin-down ages and the predictions of standard evolutionary models appear to be the result of ignoring element diffusion in such evolutionary models. Indeed, such discrepancies vanish when account is taken of diffusion.