The temperature and density dependencies of the exciton dephasing time in In 0.18 Ga 0.82 As/GaAs single quantum wells with different thicknesses have been measured by degenerate four-wave mixing. The excitonphonon scattering contribution to the dephasing is isolated by extrapolating the dephasing rate to zero-exciton density. From the temperature dependence of this rate we have deduced the linewidth broadening coefficients for acoustic and optical phonons. We find acoustic-phonon coefficients that increase from 1.6 to 3 eV/K when increasing the well width from 1 to 4 nm. This is in quantitative agreement with theoretical predictions when the spatial extension of the exciton wave function, strongly penetrating into the GaAs barrier in thin In x Ga 1Ϫx As quantum wells, is taken into account. The optical-phonon coefficient does not show a systematic dependence on well thickness, and is comparable with the value for bulk GaAs. ͓S0163-1829͑99͒02103-7͔