In this paper, an apparatus for measurements of the thermal conductivity of materials as a continuous function of the temperature is described. It works on the principle of continuous recording of the temperature difference between the surfaces of the sample caused by the flux of heat while the ambient temperature is linearily changed. The theoretical background described in the preceding paper by E. Piorkowska and A. Galeski [J. Appl. Phys.60, ▪▪▪▪ (1986)] showed that the utilization of the results of two continuous measurements with different heat fluxes makes it possible to eliminate all components of the thermal response of the sample except the one depending on the heat flux and the thermal conductivity only. Testing measurements were conducted on samples of several nonmetalic materials. The results obtained indicate high precision of the method and the apparatus and reveal new possibilities of studying the thermal conductivity of materials during phase transitions.