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Thermocurrent flowing through a single-molecule device contains valuable information about the quantum properties of the molecular structure and, in particular, on its electronic and phononic excitation spectra, and entropy. Furthermore, accessing the thermoelectric heat-to-charge conversion efficiency experimentally can help to select suitable molecules for future energy conversion devices, which - predicted by theoretical studies - could reach unprecedented efficiencies. However, one of the major challenges in quantifying thermocurrents in nanoscale devices is to determine the exact temperature bias applied to the junction. In this work, we have incorporated a superconductor-normal metal-superconductor (SNS) Josephson junction thermometer into a single-molecule device. The critical current of the Josephson junction depends accurately on minute changes of the electronic temperature in a wide temperature range from 100 mK to 1.6 K. Thus, we present a device architecture which can enable thermoelectric experiments on single molecules down to millikelvin temperatures with high precision.