We have studied the topological insulator Bi$_2$Te$_3$(111) by means of helium atom scattering. The average electron-phonon coupling $lambda$ of Bi$_2$Te$_3$(111) is determined by adapting a recently developed quantum-theoretical derivation of the helium scattering probabilities to the case of degenerate semiconductors. Based on the Debye-Waller attenuation of the elastic diffraction peaks of Bi$_2$Te$_3$(111), measured at surface temperatures between $110~mbox{K}$ and $355~mbox{K}$, we find $lambda$ to be in the range of $0.04-0.11$. This method allows to extract a correctly averaged $lambda$ and to address the discrepancy between previous studies. The relatively modest value of $lambda$ is not surprising even though some individual phonons may provide a larger electron-phonon interaction. Furthermore, the surface Debye temperature of Bi$_2$Te$_3$(111) is determined as ${rm Theta}_D = (81pm6)~mbox{K}$. The electronic surface corrugation was analysed based on close-coupling calculations. By using a corrugated Morse potential a peak-to-peak corrugation of 9% of the lattice constant is obtained.
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