Using the L3 detector, the branching ratio BR(b --> taunuX) has been measured using a sample of Z --> bbBAR events tagged by high momentum and high transverse momentum leptons in one hemisphere and with missing energy in the opposite hemisphere. From a sample of 948000 hadronic events we find BR(b --> taunuX) = (2.4 +/- 0.7 (stat.) +/- 0.8 (syst.))%.