The thermal conductivities of both the solid sulfur and the liquid sulfur were continuously measured with an accuracy of ±1.0%. The temperature range in this experiment is from 0° to 160°C. In the temperature range from 0° to 160°C, two transition points exist; one of them is the transition point from rhombic sulfur to monoclinic sulfur at 95.5°C, and the other is the change point from monoclinic sulfur to liquid sulfur at 118.9°C. At each transition point, discontinuous and conspicuous changes in thermal conductivities can be observed by this experiment.Moreover, a new result was obtained: The thermal conductivity of monoclinic sulfur decreases with increasing temperature, although the conductivity which was obtained by Kaye's experiment increases with temperature increment.