One component of the REX instrument on NASA's New Horizons spacecraft was an investigation of the radio continuum radiation from Pluto and Charon during the flyby on 14 July 2015. The planetary thermal emission was recorded at a wavelength of 4.17 cm (7.18 GHz) during approach, departure, and specifically on the non-illuminated hemispheres of Pluto and Charon during the respective intervals between occultation ingress and egress. We derive the brightness temperatures for these disk-resolved and unresolved observations. The mean values and 1σ deviations of brightness temperature for the unresolved sunlit disk are 33.2 ± 1.4 K and 47.2 ± 5.3 K for Pluto and Charon, respectively, consistent with the global albedos of the two bodies as well as with previous ground-based estimates at smaller wavelengths. A slightly colder temperature of 29.0 ± 2.5 K was determined for the disk-integrated nightside of Pluto and a larger drop in temperature was observed for Charon (40.9 ± 0.9 K), implying a smaller thermal inertia for Charon than Pluto. The measured brightness temperature of Pluto across the nightside diametric scan reached a maximum of 29.0 ± 1.5 K in the center of the disk. The profile shape is attributed to an emissivity effect, which favors thermal emission toward higher elevation angles. As a first approximation, the effective emissivity for thermal emission is calculated for the case when Pluto and Charon are uniformly smooth homogenous spheres. Under this assumption, the effective emissivity for these observations is close to unity for all probable surface constituents, implying that the effective temperature of the Pluto subsurface is only a few percent higher than the observed brightness temperature. A considerably lower subsurface emissivity is implied, however, if the higher atmospheric temperatures near the surface determined from the REX occultation measurements are also valid for the subsurface.