  We present an analysis of polarimetric observations at 22 GHz of the water
vapor masers in NGC 4258 obtained with the VLA and the GBT. We do not detect
any circular polarization in the spectrum indicative of Zeeman-induced
splitting of the maser lines of water, a non-paramagnetic molecule. We have
improved the 1-sigma upper limit estimate on the toroidal component of the
magnetic field in the circumnuclear disk of NGC 4258 at a radius of 0.2 pc from
300 mG to 90 mG. We have developed a new method for the analysis of spectra
with blended features and derive a 1-sigma upper limit of 30 mG on the radial
component of the magnetic field at a radius of 0.14 pc. Assuming thermal and
magnetic pressure balance, we estimate an upper limit on the mass accretion
rate of ~10^(-3.7) M_sun/yr for a total magnetic field of less than 130 mG. We
discuss the ramifications of our results on current maser models proposed to
explain the observed maser emission structure and the consequences for current
accretion theories. We find from our magnetic field limits that the thin-disk
model and the jet-disk model are better candidates for accounting for the
extremely low-luminosity nature of NGC 4258, than models that include
advection-dominated accretion flows.
