We present heat-transport measurements conducted with a vacuum-operated scanning thermal microscope to study the thermal conductance of monolayers of nine different alkane thiols self-assembled on Au(111) surfaces as a function of their length (2 to 18 methylene units). The molecular thermal conductance is probed in a confined area with a diameter below 10 nm in the contact between a silicon tip and the self-assembled monolayer. This yields a pWK(-1) sensitivity per molecule at a tip temperature of 200-300 °C versus the gold at room temperature. We found a conductance variance of up to a factor of 3 as a function of alkane chain length, with maximum conductance for a chain length of four carbon atoms.