  We present new white dwarf evolutionary sequences for low-metallicity
progenitors. White dwarf sequences have been derived from full evolutionary
calculations that take into account the entire history of progenitor stars,
including the thermally-pulsing and the post-asymptotic giant branch phases. We
show that for progenitor metallicities in the range 0.00003--0.001, and in the
absence of carbon enrichment due to the occurrence of a third dredge-up
episode, the resulting H envelope of the low-mass white dwarfs is thick enough
to make stable H burning the most important energy source even at low
luminosities. This has a significant impact on white dwarf cooling times. This
result is independent of the adopted mass-loss rate during the
thermally-pulsing and post-AGB phases, and the planetary nebulae stage. We
conclude that in the absence of third dredge-up episodes, a significant part of
the evolution of low-mass white dwarfs resulting from low-metallicity
progenitors is dominated by stable H burning. Our study opens the possibility
of using the observed white dwarf luminosity function of low-metallicity
globular clusters to constrain the efficiency of third dredge up episodes
during the thermally-pulsing AGB phase of low-metallicity progenitors.
