  We report VLA 1 sigma upper limits of 1.5% and 3% on the intrinsic circular
and linear fractional polarizations, respectively, of the water vapor maser
emission 0.2 pc from the central engine of NGC 4258. A corresponding 0.5% upper
limit on any Zeeman-splitting-induced circular polarization translates to a 1
sigma upper limit on the parallel, or toroidal, component of the magnetic field
of 300 mG. Assuming magnetic and thermal pressure balance in the disk, this
magnetic field upper limit corresponds to a model-dependent estimate of the
accretion rate through the molecular disk of 10^-1.9 alpha solar masses per
year for the case where the magnetic field lies along the line of sight.
