We report measurements of partial branching fractions for inclusive charmless semileptonic B decays ¯¯¯ B → X u ℓ ¯ ν and the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element | V u b | . The analysis is based on a sample of 467 × 10 6 Υ ( 4 S ) → B ¯¯¯ B decays recorded with the BABAR detector at the PEP-II e + e − storage rings. We select events in which the decay of one of the B mesons is fully reconstructed and an electron or a muon signals the semileptonic decay of the other B meson. We measure partial branching fractions Δ B in several restricted regions of phase space and determine the CKM element | V u b | based on different QCD predictions. For decays with a charged lepton momentum p ∗ ℓ > 1.0     GeV in the B meson rest frame, we obtain Δ B = ( 1.80 ± 0.13 stat ± 0.15 sys ± 0.02 theo ) × 10 − 3 from a fit to the two-dimensional M X − q 2 distribution. Here, M X refers to the invariant mass of the final state hadron X and q 2 is the invariant mass squared of the charged lepton and neutrino. From this measurement we extract | V u b | = ( 4.33 ± 0.24 exp  ± 0.15 theo ) × 10 − 3 as the arithmetic average of four results obtained from four different QCD predictions of the partial rate. We separately determine partial branching fractions for ¯¯¯ B 0 and B − decays and derive a limit on the isospin breaking in ¯¯¯ B → X u ℓ ¯ ν decays.