We present a determination of the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element |Vcb| using the decay B→Dℓνℓ (ℓ=e,μ) based on 711 fb-1 of e+e-→ (4S) data recorded by the Belle detector and containing 772×106 BB¯ pairs. One B meson in the event is fully reconstructed in a hadronic decay mode, while the other, on the signal side, is partially reconstructed from a charged lepton and either a D+ or D0 meson in a total of 23 hadronic decay modes. The isospin-averaged branching fraction of the decay B→Dℓνℓ is found to be B(B0→D-ℓ+νℓ)=(2.31±0.03(stat)±0.11(syst))%. Analyzing the differential decay rate as a function of the hadronic recoil with the parametrization of Caprini, Lellouch, and Neubert and using the form-factor prediction G(1)=1.0541±0.0083 calculated by FNAL/MILC, we obtain ηEW|Vcb|=(40.12±1.34)×10-3, where ηEW is the electroweak correction factor. Alternatively, assuming the model-independent form-factor parametrization of Boyd, Grinstein, and Lebed and using lattice QCD data from the FNAL/MILC and HPQCD collaborations, we find ηEW|Vcb|=(41.10±1.14)×10-3.