A measurement of the ratio of the branching fractions, R$_{τ/e}$ = B(W → τν)/B(W → eν), is performed using a sample of W bosons originating from top-quark decays to final states containing τ-leptons or electrons. This measurement uses pp collisions at $ \sqrt{s} $ = 13 TeV, collected by the ATLAS experiment at the Large Hadron Collider during Run 2, corresponding to an integrated luminosity of 140 fb$^{−1}$. The W → τν$_{τ}$ (with τ → eν$_{e}$ν$_{τ}$) and W → eν$_{e}$ decays are distinguished using the differences in the impact parameter distributions and transverse momentum spectra of the electrons. The measured ratio of branching fractions R$_{τ/e}$ = 0.975 ± 0.012 (stat.) ± 0.020 (syst.), is consistent with the Standard Model assumption of lepton flavour universality in W-boson decays.[graphic not available: see fulltext]