We present a search for the rare leptonic decays B+ -> e(+)nu(e) and B+ -> mu(+) nu(mu) using the full Upsilon(4S) data sample of 772 x 10(6) B (B) over bar pairs collected with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider. One of the B mesons from the Upsilon(4S) -> B (B) over bar 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 beta(B+ -> e(+)nu(e)) < 3.5 x 10(-6) and beta(B+ -> mu(+)nu(mu)) < 2.7 x 10(-6) at 90% confidence level.