  The present work is designed to explore the evolution of helium-core white
dwarf (HeWD) stars for the case of metallicities much lower than the solar one
(Z=0.001 and Z=0.0002). Evolution is followed in a self-consistent way with the
predictions of detalied and new non-grey atmospheres, time-dependent element
diffusion and the history of the white dwarf progenitor. Reliable initial
models for low mass HeWDs are obtained by applying mass loss rates to a 1msun
stellar model. The loss of angular momentum caused by gravitational wave
emission and magnetic stellar wind braking are considered. Model atmospheres,
based on a detailed treatment of the microphysics entering the WD atmosphere
enable us to provide accurate colours and magnitudes at both early and advanced
evolutionary stages. We find that most of our evolutionary sequences experience
several episodes of hydrogen thermonuclear flashes. In particular, the lower
the metallicity, the larger the minimum stellar mass for the occurrence fo
flashes induced by CNO cycle reactions. The existence of a mass-threshold for
the occurrence of diffusion-induced CNO flashes leadss to a marked dichotomy in
the age of our models. Another finding of this study is that our HeWD models
experience unstable hydrogen burning via PP nuclear reactions at late cooling
stages as a result of hydrogen chemically diffusing inwards. Such PP flashes
take place in models with very low metal content. We also find that models
experiencing CNO flashes exhibit a pronouncede turn-off in most of their
colours at M_V=16 approximately. Finally, colour-magnitude diagrams for our
models are presented and compared with recent observational data of HeWD
candidates in the globular clusters NGC 6397 and 47 Tucanae.
