The thermal conductivity of polymethyl methacrylate (PMMA) and borosilicate crown glass BK7 has been studied. The transient hot-wire technique has been employed, and measurements cover a temperature range from room temperature up to 350 K for PMMA and up to 500 K for BK7. The technique is applied here in a novel way that minimizes all remaining thermal-contact resistances. This allows the apparatus to operate in an absolute way and with very low uncertainty. The method makes use of a soft silicone paste material between the hot wires and the solid under test. Measurements of the transient temperature rise of the wires in response to an electrical heating step over a period of 20 μs up to 5 s allow an absolute determination of the thermal conductivity of the solid, as well as of the silicone paste. The method is based on a full theoretical model with equations solved by a two-dimensional finite-element method applied to the exact geometry. At the 95% confidence level, the standard deviations of the thermal conductivity measurements are 0.09% for PMMA and 0.16% for BK7, whereas the standard uncertainty of the technique is less than 1.5%.