The ratio of branching ratios of the W boson to muons and electrons, R-W(mu/e) = B(W -> mu nu)/B(W -> e nu), has been measured using 140 fb(-1) of pp collision data at root s = 13TeV collected with the ATLAS detector at the LHC, probing the universality of lepton couplings. The ratio is obtained from measurements of the t (t) over bar 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-Z(mu mu/ee) for the Z boson measured in inclusive Z -> ee and Z -> mu mu events. By using the precise value of R-Z(mu mu/ee) determined from e(+)e(-) colliders, the ratio R-W(mu/e) is determined to be R-W(mu/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-Z(mu mu/ee), 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.