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We have measured the exciton dephasing time in I n A s/G a A s quantum dot molecules having different interdot barrier thicknesses in the temperature range from 5 to 60 K, using a highly sensitive four-wave mixing heterodyne technique. At 5 K dephasing times of several hundred picoseconds are found. Moreover, a systematic dependence of the dephasing dynamics on the barrier thickness is observed. These results show how the quantum-mechanical coupling of the electronic wave functions in the molecules affects both the exciton radiative lifetime and the exciton-acoustic phonon interaction.