We introduce a new technique for measurement of the thermal conductivity of ultrathin films of single-walled carbon nanotubes (SWNTs) utilizing IR radiation as heat source and the SWNT film as thermometer. The technique is applied to study the temperature dependence of the thermal conductivity of an as-prepared SWNT film obtained in the electric arc discharge process and a film of purified SWNTs prepared by vacuum filtration. The interplay between thermal and electrical transport in SWNT networks is analyzed in relation to the type of intertube junctions and the possibility of optimizing the thermal and electrical properties of SWNT networks for specific applications is discussed.