  Context: About 10% of white dwarfs have magnetic fields with strength in the
range between about 10^5 and 3x10^8 G. It is not known whether the remaining
white dwarfs are not magnetic, or if they have a magnetic field too weak to be
detected with the techniques adopted in the large surveys. Aims. We describe
the results of the first survey specifically devised to clarify the detection
frequency of kG-level magnetic fields in cool DA white dwarfs. Methods: Using
the FORS1 instrument of the ESO VLT, we have obtained Balmer line circular
spectropolarimetric measurements of a small sample of cool (DA6 - DA8) white
dwarfs. Using FORS and UVES archive data, we have also revised numerous white
dwarf field measurements previously published in the literature. Results: We
have discovered an apparently constant longitudinal magnetic field of \sim9.5
kG in the DA6 white dwarf WD2105-820. This star is the first weak-field white
dwarf that has been observed sufficiently to roughly determine the
characteristics of its field. The available data are consistent with a simple
dipolar morphology with magnetic axis nearly parallel to the rotation axis, and
a polar strength of \simeq 56 kG. Our re-evaluation of the FORS archive data
for white dwarfs indicates that longitudinal magnetic fields weaker than 10 kG
had previously been correctly identified in at least three white dwarfs.
Conclusions: We find that the probability of detecting a field of kG strength
in a DA white dwarf is of the order of 10% for each of the cool and hot DA
stars. If there is a lower cutoff to field strength in white dwarfs, or a field
below which all white dwarfs are magnetic, the current precision of
measurements is not yet sufficient to reveal it.
