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We have measured the thermal conductivity of two diamond films grown by a chemical vapor deposition process. At room temperature the thermal conductivity is of the order of 10 W cm-1 K-1, i.e., as high as naturally occurring (type Ia) single‐crystal diamonds. This value exceeds the thermal conductivity of copper at room temperature by a factor of 2. Since these films consist of microcrystallites of diamonds of dimensions on the order of 2 μm, boundary scattering of phonons is expected to be large. The expected effect of boundary scattering on the lattice thermal conductivity is calculated, and is in qualitative agreement with the observed results above about 30 K. However, important differences between the measured conductivity and that expected for boundary scattering are observed below this temperature. It is proposed that a small amount of disorder present in the lattice (identified through Raman studies) can account for this unusual behavior.