We report a measurement of the branching fraction of the exclusive charmless semileptonic decay B + → ω ℓ + ν , where ℓ is either an electron or a muon. We use samples of B + mesons tagged by a reconstructed charmed semileptonic decay of the other B meson in the event. The measurement is based on a data set of 426.1     fb − 1 of e + e − collisions at a center-of-mass energy of 10.58 GeV recorded with the BABAR detector at the PEP-II asymmetric-energy e + e − storage rings. We measure a branching fraction of B ( B + → ω ℓ + ν ) = ( 1.35 ± 0.21 ± 0.11 ) × 10 − 4 , where the uncertainties are statistical and systematic, respectively. We also present measurements of the partial branching fractions in three bins of q 2 , the invariant-mass squared of the lepton-neutrino system, and we compare them to theoretical predictions of the form factors.