  We present a grid of evolutionary tracks for low-mass white dwarfs with
helium cores in the mass range from 0.179 to 0.414 Msol. The lower mass limit
is well-suited for comparison with white dwarf companions of millisecond
pulsars. The tracks are based on a 1 Msol model sequence extending from the
pre-main sequence stage up to the tip of the red-giant branch. Applying large
mass loss rates at appropriate positions forced the models to move off the
giant branch. The further evolution was then followed across the
Hertzsprung-Russell diagram and down the cooling branch. At maximum effective
temperature the envelope masses above the helium cores increase from 0.6 to 5.4
x 10^{-3} Msol for decreasing mass. We carefully checked for the occurrence of
thermal instabilities of the hydrogen shell by adjusting the computational time
steps accordingly. Hydrogen flashes have been found to take place only in the
mass interval 0.21 < M/Msol < 0.3. The models show that hydrogen shell burning
contributes significantly to the luminosity budget of white dwarfs with helium
cores. For very low masses the hydrogen shell luminosity remains to be dominant
even down to effective temperatures well below 10000K. Accordingly, the
corresponding cooling ages are significantly larger than those gained from
model calculations which neglect nuclear burning or the white dwarf progenitor
evolution. Using the atmospheric parameters of the white dwarf in the PSR
J1012+5307 system we determined a mass of M=0.19 +/- 0.02 Msol and a cooling
age of 6 +/- 1 Gyr, in good agreement with the spin-down age, 7 Gyr, of the
pulsar.
