For the first time in an extragalactic source, we detect linearly polarized H2O maser emission associated with the molecular accretion disk of NGC 1068. The position angles of the electric polarization vectors are perpendicular to the axes of filamentary structures in the molecular accretion disk. The inferred magnetic field threading the molecular disk must lie within ∼35° of the sky plane. The orientation of the magnetic fields relative to the disk plane implies that the maser region is unstable to hydromagnetically powered outflow; we speculate that the maser region may be the source of the larger-scale molecular outflow found in Atacama Large Millimeter/submillimeter Array studies. The new very long baseline interferometry observations also reveal a compact radio continuum source, NGC 1068*, aligned with the near-systemic maser spots. The molecular accretion disk must be viewed nearly edge on, and the revised central mass is M = (16.6 ± 0.1) × 106 M ☉.