  We present a search for the rare leptonic decays $B^+ \to e^+ \nu_e$ and $B^+
\to \mu^+ \nu_\mu$, using the full $\Upsilon(4S)$ data sample of $772\times
10^6$ $B\bar{B}$ pairs collected with the Belle detector at the $\rm {\small
KEKB}$ asymmetric-energy $e^+ e^-$ collider. One of the $B$ mesons from the
$\Upsilon(4S) \to B\bar{B}$ decay is fully reconstructed in a hadronic mode,
while the recoiling side is analyzed for the signal decay. We find no evidence
of a signal in any of the decay modes. Upper limits of the corresponding
branching fractions are determined as ${\cal B}(B^+ \to e^+ \nu_e)<3.5\times
10^{-6}$ and ${\cal B}(B^+ \to \mu^+ \nu_\mu)<2.7 \times 10^{-6}$ at 90$\%$
confidence level.
