  Searches for magnetic fields in white dwarfs have clarified both the
frequency of occurrence and the global structure of the fields found down to
field strengths of the order of 500 kG. Below this level, the situation is
still very unclear. We are studying the weakest fields found in white dwarfs to
determine the frequency of such fields and their structure. We describe a very
sensitive new method of measuring such fields in DA (H-rich) white dwarfs, and
search for a field in the brightest such star, 40 Eri B. Our new method makes
use of the strongly enhanced polarisation signal in the sharp core of Halpha.
We find that with one-hour integrations with the high-resolution
spectropolarimeter ESPaDOnS on the 3.6-m CFHT, we can reach a standard error fo
the longitudinal field of about 85 G, the smallest error ever achieved for any
white dwarf. Nevertheless, we do not detect a magnetic field in this star.
Observations with ISIS at the WHT, and the Main Stellar Spectrograph at the
SAO, support the absence of a field at somewhat lower precision. The new method
is very efficient; it is shown that for suitable DA stars the integration time,
with ESPaDOnS on a 3.6-m telescope, to reach a 500 G standard error on a white
dwarf of V = 12.5, is about half an hour, about the same as the time required
on an ESO 8-m telescope with FORS using conventional low-resolution
spectropolarimetry.
