Engineering lattice thermal conductivity requires to control the heat carried by atomic vibration waves, the phonons. The key parameter for quantifying it is the phonon lifetime, limiting the travelling distance, whose determination is however at the limits of instrumental capabilities. Here, we show the achievement of a direct quantitative measurement of phonon lifetimes in a single crystal of the clathrate Ba7.81Ge40.67Au5.33, renowned for its puzzling 'glass-like' thermal conductivity. Surprisingly, thermal transport is dominated by acoustic phonons with long lifetimes, travelling over distances of 10 to 100 nm as their wave-vector goes from 0.3 to 0.1 Å-1. Considering only low-energy acoustic phonons, and their observed lifetime, leads to a calculated thermal conductivity very close to the experimental one. Our results challenge the current picture of thermal transport in clathrates, underlining the inability of state-of-the-art simulations to reproduce the experimental data, thus representing a crucial experimental input for theoretical developments.Phonon lifetime is a fundamental parameter of thermal transport however its determination is challenging. Using inelastic neutron scattering and the neutron resonant spin-echo technique, Lory et al. determine the acoustic phonon lifetime in a single crystal of clathrate Ba7.81Ge40.67Au5.33.