The absence of inversion symmetry leads to a strong spin-orbit splitting of the upper valence band of semiconducting single-layer transition-metal dichalchogenides such as MoS2 or WS2. This permits a direct comparison of the electron-phonon coupling strength in states that only differ by their spin. Here, the electron-phonon coupling in the valence band maximum of single-layer WS2 is studied by first-principles calculations and angle-resolved photoemission. The coupling strength is found to be drastically different for the two spin-split branches, with calculated values of λK=0.0021 and 0.40 for the upper and lower spin-split valence band of the freestanding layer, respectively. This difference is somewhat reduced when including scattering processes involving the Au(111) substrate present in the experiment but it remains significant, in good agreement with the experimental results.