Faraday rotation observations of polarized radiation from natural radio sources are unique among remote diagnostics of the solar corona in that they provide information on the coronal magnetic field. Dual frequency radio polarization measurements yield the rotation measure, a quantity that is proportional to the integral along the line of sight of the product of the electron density and the line-of-sight component of the magnetic field. We made linear polarization observations with the NRAO Very Large Array of 13 polarized radio sources occulted by the solar corona. The observations were made at frequencies of 1465 and 1665 MHz on four days in 1997 May and cover a 20 day period, sampling elongations ranging from about 5 to 14 Rsolar. The magnitudes of the rotation measures observed range from about 11 to 0 rad m-2. The relatively low values for the rotation measures are due to the solar minimum configuration of the corona at the time of the observations, with the lines of sight to the sources generally not crossing sector boundaries. The largest rotation measure was observed for the extended radio source 3C 79 on 1997 May 11 and corresponds to a case in which the line of sight passed next to the streamer belt at small solar elongations. We have developed a three-dimensional model of the solar corona that is in excellent agreement with the observed rotation measures, as well as being completely consistent with other coronal diagnostics such as coronagraph images. In particular, our observations support the coronal magnetic field model of Pätzold et al. (1987) they would be inconsistent with coronal magnetic fields significantly weaker or stronger than this model. The plasma density distribution in the corona is successfully modeled by a dense streamer belt component and a more tenuous coronal hole component. Details of these models are given in § 3 of this paper. The principal disagreement between the model and observations occurs for three lines of sight for which the model predicts nearly zero rotation measure but for which we measure small but significant values of -1 to -2 rad m-2. These lines of sight passed over the solar polar regions. We discuss the possibility that these residual rotation measures are due to static coronal plasma structures, not described by global coronal models, or to very long wavelength coronal Alfvén waves. Fluctuations in the rotation measure on timescales of a few hours were observed for some sources and not others. When detected, they were of order 1-2 rad m-2 and occurred on timescales of several hours.