A sample of 25000 Z(0) --> tau(-)tau(+) events collected by the DELPHI experiment at LEP in 1991 and 1992 is used to measure the leptonic branching fractions of the tau lepton. The results are B(tau --> e nu<(nu)over bar>) = (17.51+/-0.39)% and B(tau --> mu nu<(nu)over bar>) = (17.02+/-0.31)%. The ratio of the muon and electron couplings to the weak charged current is measured to be g(mu)/g(e) = 1.000+/-0.013, satisfying e-mu universality. The average leptonic branching fraction corrected to the value for a massless lepton, assuming e-mu universality, is found to be B(tau --> l nu<(nu)over bar>) = (17.50+/-0.25)%.
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