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