The thermal conductivity of a type I diamond has been measured between 2 degrees K and room temperature. Above 100 degrees K the temperature variation of conductivity accords with Peierls's theory of Umklapp processes. To account for the results at the lowest temperatures it is necessary to assume that some specular reflexion of phonons occurs at the crystal boundaries. Casimir's theoretical treatment of boundary scattering has been extended to take into account the finite length of specimens and the smoothness of the crystal walls, and the experimental results show that at 3 degrees K the proportion of specular reflexions is as great as 40%. Between 50 and 100 degrees K the conductivity is considerably less than that calculated for an ideal diamond, and some conclusions are drawn as to the imperfections which cause this discrepancy.