In support of our ongoing program to measure coronal magnetic fields using Faraday rotation (the rotation of the plane of polarization when linearly polarized light propagates through a magnetized plasma), we report on measurements of angular broadening of cosmic radio sources as they were occulted by the corona. Angular broadening results from plasma density irregularities elongated along the coronal magnetic field that cause radio wave scattering. In radioastronomical observations, the measured intensity (power per unit area per unit solid angle along the path to the detector) is the convolution of the true radio source intensity with a point spread function (PSF, the effective response of an imaging system to a point-like source). In most radio interferometric applications, the PSF is simply the synthesized beam; however, when observing through a turbulent plasma like the corona, the PSF is the convolution of the synthesized beam with the power pattern of the angular broadening. The angular broadening acts to reduce the measured intensity of a radio source. This can have important consequences for coronal Faraday rotation studies because the error associated with measuring the polarization angle is inversely proportional to the intensity of the linearly polarized light. We made full-polarization observations at 1 – 2 GHz frequencies using the Karl G. Jansky Very Large Array of 21 linearly polarized cosmic radio sources occulted by the solar corona in July and August, 2015. The radio sources were scattered over a range of heliocentric distances; however, sources within a heliocentric distance of 10 solar radii experienced the most pronounced angular broadening. The observed angular broadening is consistent with a 15% - 35% decrease in intensity. We discuss the implications of these angular broadening measurements on the observed coronal Faraday rotation as well as potential methods to correct for this phenomenon in mapping the radio sources.