  White dwarf atmospheres are subjected to gravitational potentials around
$10^5$ times larger than occur on Earth. They provide a unique environment in
which to search for any possible variation in fundamental physics in the
presence of strong gravitational fields. However, a sufficiently strong
magnetic field will alter absorption line profiles and introduce additional
uncertainties in measurements of the fine structure constant. Estimating the
magnetic field strength is thus essential in this context. Here we model the
absorption profiles of a large number of atomic transitions in the white dwarf
photosphere, including first-order Zeeman effects in the line profiles, varying
the magnetic field as a free parameter. We apply the method to a high
signal-to-noise, high-resolution, far-ultraviolet HST/STIS spectrum of the
white dwarf G191-B2B. The method yields a sensitive upper limit on its magnetic
field of $B < 2300$ Gauss at the $3\sigma$ level. Using this upper limit we
find that the potential impact of quadratic Zeeman shifts on measurements of
the fine structure constant in G191-B2B is 4 orders of magnitude below
laboratory wavelength uncertainties.
