The properties of a transiting planet's host star are written in its transit light curve. The light curve can reveal the stellar density ({rho}_*_) and the limb-darkening profile in addition to the characteristics of the planet and its orbit. For planets with strong prior constraints on orbital eccentricity, we may measure these stellar properties directly from the light curve; this method promises to aid greatly in the characterization of transiting planet host stars targeted by the upcoming NASA Transiting Exoplanet Survey Satellite mission and any long-period, singly transiting planets discovered in the same systems. Using Bayesian inference, we fit a transit model, including a nonlinear limb-darkening law, to 66 Kepler transiting planet hosts to measure their stellar properties. We present posterior distributions of {rho}*, limb-darkening coefficients, and other system parameters for these stars. We measure densities to within 5% for the majority of our target stars, with the dominant precision-limiting factor being the signal-to-noise ratio of the transits. Of our measured stellar densities, 95% are in 3{sigma} or better agreement with previously published literature values. We make posterior distributions for all of our target Kepler objects of interest available online at 10.5281/zenodo.1028515.