We have performed thermal-conductance measurements on polished single crystals of pure silicon in the low-temperature boundary-scattering regime. Our data show that the thermal conductance depends not only on the sample size, but also on the size and spacing of the thermometers used to measure the temperature gradient along the crystal. We have analyzed the heat transport in terms of phonon blackbody radiation subject to surface scattering from the attached thermometers and from the free surfaces of the crystal. We have made this analysis quantitative by using Monte Carlo techniques, and have calculated the magnitude of both surface scattering effects. We show that thermal-conductance measurements provide a very sensitive and precise technique for studying diffuse phonon scattering at crystal surfaces, and find that our highly polished and clean silicon surfaces will specularly reflect more than 99% of the incident phonons below 1 K, temperatures which correspond to dominant phonon frequencies less than 90 GHz.