We present an analysis of exclusive charmless semileptonic B-meson decays based on 83×106 BB̄ pairs recorded with the BABAR detector at the Υ(4S) resonance. Using isospin symmetry, we measure branching fractions B(B0→π- +ν)=(1.38±0.10±0.16±0.08)×10-4 and B(B0→ρ- +ν)=(2.14±0.21±0.48±0.28)×10-4, where the errors are statistical, experimental systematic, and due to form-factor shape uncertainties. We compare the measured distribution in q2, the momentum-transfer squared, with theoretical predictions for the form factors from lattice QCD and light-cone sum rules, and extract the Cabibbo-Kobayashi- Maskawa (CKM) matrix element |Vub|=(3.82±0.14±0.22±0.11-0. 52+0.88)×10-3 from B→π ν, where the fourth error reflects the uncertainty of the form-factor normalization. © 2005 The American Physical Society.