We present observations of the 22 GHz water megamasers in the circumnuclear disk of NGC 3079 obtained with the Green Bank Telescope. The data are analyzed for circular polarization due to the Zeeman-induced splitting of the water maser lines. No circular polarization is detected and we derive a 1 sigma upper limit of 11 mG for the toroidal magnetic field at ~0.64 pc from the central black hole. This is the tightest upper limit for the magnetic field around a black hole to date. We use the magnetic field limit to derive an estimate of the mass accretion onto the central black hole. In addition to the polarimetric results, we present an observation of rapid variability in the maser lines, which we explain as weak interstellar scintillation. From the scintillation parameters, we estimate an intrinsic size of the mostly saturated maser features of ~12 microarcseconds. This is consistent with models assuming a thick, clumpy accretion disk.