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The thermal conductivity $K$ of single crystals of silicon has been measured from 3 to 1580\ifmmode^\circ\else\textdegree\fi{}K and of single crystals of germanium from 3 to 1190\ifmmode^\circ\else\textdegree\fi{}K. These measurements have been made using a steady-state, radial heat flow apparatus for $T>300\ifmmode^\circ\else\textdegree\fi{}$K and a steady-state, longitudinal flow apparatus for $T<300\ifmmode^\circ\else\textdegree\fi{}$K to give absolute $K$ values. This radial flow technique eliminates thermal radiation losses at high temperatures. The accuracy of both the low-temperature apparatus and the high-temperature apparatus is approximately \ifmmode\pm\else\textpm\fi{}5%. Some special experimental techniques in using the high-temperature apparatus are briefly considered. At all temperatures the major contribution to $K$ in Si and Ge is produced by phonons. The phonon thermal conductivity has been calculated from a combination of the relaxation times for boundary, isotope, three-phonon, and four-phonon scattering, and was found to agree with the experimental measurements. Above 700\ifmmode^\circ\else\textdegree\fi{}K for Ge and 1000\ifmmode^\circ\else\textdegree\fi{}K for Si an electronic contribution to $K$ occurs, which agrees quite well with the theoretical estimates. At the respective melting points of Si and Ge, electrons and holes are responsible for 40% of the total $K$ and phonons are responsible for 60%. The measured electronic $K$ yields values for the thermal band gap at the melting point of 0.6\ifmmode\pm\else\textpm\fi{}0.1 eV for Si and 0.26\ifmmode\pm\else\textpm\fi{}0.08 eV for Ge.