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We report on a search for the rare decay modes $B^+\rightarrow e^+ \nu_e$ and
$B^+\rightarrow \mu^+ \nu_\mu$
with data collected from the BaBar detector
at the PEP-II $e^+e^-$ storage ring. This search
utilizes a new technique in which we fully reconstruct the accompanying
$B^-$ in $\Upsilon(4S)\righarrow B^+ B^-$ events, and look for a mono-energetic
lepton in the $B^+$ rest frame. No signal candidates are observed in either
of the channels, consistent with the expected background,
in a data sample of approximately 229 million $B\bar{B}$ pairs. The branching-fraction upper limits are set
at $\mathcal{B}(B^+\rightarrow e^+\nu_{e})<7.9 \times 10^{-6}$ and $\mathcal{B}(B^+\rightarrow\mu^+\nu_{\mu})<6.2 \times 10^{-6}$
at the 90\% confidence level.
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