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We present an analysis of exclusive charmless semileptonic $B$-meson decays based on $377\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs recorded with the BABAR detector at the $\mathit{\ensuremath{\Upsilon}}(4S)$ resonance. We select four event samples corresponding to the decay modes ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, and ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$ and find the measured branching fractions to be consistent with isospin symmetry. Assuming isospin symmetry, we combine the two $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\ell}\ensuremath{\nu}$ samples, and similarly the two $B\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\ell}\ensuremath{\nu}$ samples, and measure the branching fractions $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu})=(1.41\ifmmode\pm\else\textpm\fi{}0.05\ifmmode\pm\else\textpm\fi{}0.07)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ and $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu})=(1.75\ifmmode\pm\else\textpm\fi{}0.15\ifmmode\pm\else\textpm\fi{}0.27)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, where the errors are statistical and systematic. We compare the measured distribution in ${q}^{2}$, 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 $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\ell}\ensuremath{\nu}$ in the range ${q}^{2}<12\text{ }\text{ }{\mathrm{GeV}}^{2}$ and the most recent QCD light-cone sum-rule calculations, we obtain $|{V}_{ub}|=(3.78\ifmmode\pm\else\textpm\fi{}{0.13}_{\ensuremath{-}0.40}^{+0.55})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, where the errors refer to the experimental and theoretical uncertainties. From a simultaneous fit to the data over the full ${q}^{2}$ range and the FNAL/MILC lattice QCD results, we obtain $|{V}_{ub}|=(2.95\ifmmode\pm\else\textpm\fi{}0.31)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ from $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\ell}\ensuremath{\nu}$, where the error is the combined experimental and theoretical uncertainty.

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