We present a search for the decay B^+→ℓ^+ν_ℓ(ℓ=τ, μ, or e) in (458.9±5.1)×10^6 BB̅ pairs recorded with the BABAR detector at the PEP-II B-factory. We search for these B decays in a sample of B^+B^- events where one B-meson is reconstructed as B^-→D^0ℓ^-ν̅ X. Using the method of Feldman and Cousins, we obtain B(B^+→τ^+ν_τ)=(1.7±0.8±0.2)×10^(-4), which excludes zero at 2.3σ. We interpret the central value in the context of the standard model and find the B meson decay constant to be f_B^2=(62±31)×10^3 MeV^2. We find no evidence for B^+→e^+ν_e and B^+→μ^+ν_μ and set upper limits at the 90% C.L. B(B^+→e^+ν_e)<0.8×10^(-5) and B(B^+→μ^+ν_μ)<1.1×10^(-5)
