Molecular junctions exhibit a rich and tunable set of thermal transport phenomena. However, the predicted high thermoelectric efficiencies, phonon quantum interference effects, rectification, and nonlinear heat transport properties of organic molecules are yet to be verified because suitable experimental techniques have been missing. Here, by combining the break junction technique with suspended heat-flux sensors with picowatt per Kelvin sensitivity, we measured the thermal and electrical conductance of single organic molecules at room temperature simultaneously. We used this method to study the thermal transport properties of two model systems, namely, dithiol-oligo(phenylene ethynylene) and octane dithiol junctions with gold electrodes. In agreement with our density functional theory and phase-coherent transport calculations, we show that heat transport across these systems is governed by the phonon mismatch between the molecules and the metallic electrodes. This work represents the first measurement of thermal transport through single molecules and opens new opportunities for studying heat management at the nanoscale level.