The ratio of branching ratios of the W boson to muons and electrons, $$R^{\,\mu /e}_W={{\mathcal {B}}(W\rightarrow \mu \nu )}$$ R W μ / e = B ( W → μ ν ) /$${{\mathcal {B}}(W\rightarrow e\nu )}$$ B ( W → e ν ) , has been measured using $$140\,\text{ fb}^{-1}\,$$ 140 fb - 1 of pp collision data at $$\sqrt{s}=13$$ s = 13 $$\text {T}\text {e}\hspace{-1.00006pt}\text {V}$$ Te V collected with the ATLAS detector at the LHC, probing the universality of lepton couplings. The ratio is obtained from measurements of the $$t\bar{t}$$ t t ¯ production cross-section in the ee, $$e\mu $$ e μ and $$\mu \mu $$ μ μ dilepton final states. To reduce systematic uncertainties, it is normalised by the square root of the corresponding ratio $$R^{\,\mu \mu /ee}_Z$$ R Z μ μ / e e for the Z boson measured in inclusive $$Z\rightarrow ee$$ Z → e e and $$Z\rightarrow \mu \mu $$ Z → μ μ events. By using the precise value of $$R^{\,\mu \mu /ee}_Z$$ R Z μ μ / e e determined from $$e^+e^-$$ e + e - colliders, the ratio $$R^{\,\mu /e}_W$$ R W μ / e is determined to be $$\begin{aligned} R^{\,\mu /e}_W&= 0.9995\pm 0.0022\,\mathrm {(stat)}\,\pm 0.0036\,\mathrm {(syst)}\\ &\quad \pm 0.0014\,\mathrm {(ext)} . \end{aligned}$$ R W μ / e = 0.9995 ± 0.0022 ( stat ) ± 0.0036 ( syst ) ± 0.0014 ( ext ) . The three uncertainties correspond to data statistics, experimental systematics and the external measurement of $$R^{\,\mu \mu /ee}_Z$$ R Z μ μ / e e , giving a total uncertainty of 0.0045, and confirming the Standard Model assumption of lepton flavour universality in W-boson decays at the 0.5% level.