Although diamond has the highest known room temperature thermal conductivity, k∼2200W/mK, highly sp3 amorphous carbon films have k≪15W/mK. We carry out an integrated experimental and simulation study of thermal transport in ultrananocrystalline diamond (UNCD) films. The experiments show that UNCD films with a grain size of 3–5nm have thermal conductivities as high as k=12W/mK at room temperature, comparable with that of the most conductive amorphous diamond films. This value corresponds to a grain boundary (Kapitza) conductance greater than 3000MW/m2K, which is ten times larger than that previously seen in any material. Our simulations of both UNCD and individual diamond grain boundaries yield values for the grain boundary conductance consistent with the experimentally obtained value, leading us to conclude that thermal transport in UNCD is controlled by the intrinsic properties of the grain boundaries.