We present an analysis of exclusive charmless semileptonic B-meson decays based on 377×106 BB̄ pairs recorded with the B A BAR detector at the Υ(4S) resonance. We select four event samples corresponding to the decay modes B0→π-ℓ+ν, B +→π0ℓ+ν, B0→ρ -ℓ+ν, and B+→ρ0ℓ +ν and find the measured branching fractions to be consistent with isospin symmetry. Assuming isospin symmetry, we combine the two B→πℓν samples, and similarly the two B→ρℓν samples, and measure the branching fractions B(B0→π -ℓ+ν)=(1.41±0.05±0.07)×10 -4 and B(B0→ρ-ℓ+ν)=(1. 75±0.15±0.27)×10-4, where the errors are statistical and systematic. We compare the measured distribution in q2, the momentum transfer squared, with predictions for the form factors from QCD calculations and determine the Cabibbo-Kobayashi-Maskawa matrix element |V ub|. Based on the measured partial branching fraction for B→πℓν in the range q2<12GeV2 and the most recent QCD light-cone sum-rule calculations, we obtain |Vub|=(3.78±0.13- 0.40+0.55)×10-3, where the errors refer to the experimental and theoretical uncertainties. From a simultaneous fit to the data over the full q2 range and the FNAL/MILC lattice QCD results, we obtain |Vub|=(2. 95±0.31)×10-3 from B→πℓν, where the error is the combined experimental and theoretical uncertainty. © 2011 American Physical Society.