Context : About 10 % of white dwarfs have magnetic fields with strength in the range between about 10 ^ { 5 } and 5 10 ^ { 8 } G. It is not known whether the remaining white dwarfs are not magnetic , or if they have magnetic fields too weak to be detected with the techniques adopted in the large surveys .
Information is particularly lacking for the cooler ( and generally fainter ) white dwarfs .

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 \sim 9.5 kG in the DA6 white dwarf WD 2105 $ - $ 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 have previously been correctly identified in at least three white dwarfs .
However , for one of these three weak-field stars ( WD 2359 $ - $ 434 ) , UVES archive data show a \sim 100 kG mean field modulus .
Either at the time of the FORS observations the star ’ s magnetic field axis was nearly perpendicular to the line of sight , or the star ’ s magnetic field has rather complex structure .

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 .