We report a measurement of the inclusive charmless semileptonic branching fraction of B mesons in a sample of 89 x 10(6) (-)BB events recorded with the BABAR detector at the Upsilon(4S) resonance. Events are selected by fully reconstructing the decay of one B meson and identifying a charged lepton from the decay of the other B meson. The number of signal events is extracted from the mass distribution of the hadronic system accompanying the lepton and is used to determine the ratio of branching fractions B((-)B-->X(u)lnu;)/B((-)B-->Xlnu;)=[2.06+/-0.25(stat)+/-0.23(syst)+/-0.36(theo)]x10(-2). Using the measured branching fraction for inclusive semileptonic B decays, we find B((-)B-->X(u)lnu;)=[2.24+/-0.27(stat)+/-0.26(syst)+/-0.39(theo)]x10(-3) and derive the Cabibbo-Kobayashi-Maskawa matrix element |V(ub)|=[4.62+/-0.28(stat)+/-0.27(syst)+/-0.48(theo)]x10(-3).