We study heating and heat dissipation of a single c60 molecule in the junction of a scanning tunneling microscope (STM) by measuring the electron current required to thermally decompose the fullerene cage. The power for decomposition varies with electron energy and reflects the molecular resonance structure. When the STM tip contacts the fullerene the molecule can sustain much larger currents. Transport simulations explain these effects by molecular heating due to resonant electron-phonon coupling and molecular cooling by vibrational decay into the tip upon contact formation.