We present a study of B decays into semileptonic final states containing charged and neutral D1(2420) and D_{2};{*}(2460). The analysis is based on a data sample of 208 fb;{-1} collected at the Upsilon(4S) resonance with the BABAR detector at the PEP-II asymmetric-energy B factory at SLAC. With a simultaneous fit to four different decay chains, the semileptonic branching fractions are extracted from measurements of the mass difference Deltam=m(D;{**})-m(D) distributions. Product branching fractions are determined to be B(B;{+}-->D_{1};{0}l;{+}nu_{l})xB(D_{1};{0}-->D;{*+}pi;{-})=(2.97+/-0.17+/-0.17)x10;{-3}, B(B;{+}-->D_{2};{*0}l;{+}nu_{l})xB(D_{2};{*0}-->D;{(*)+}pi;{-})=(2.29+/-0.23+/-0.21)x10;{-3}, B(B;{0}-->D_{1};{-}l;{+}nu_{l})xB(D_{1};{-}-->D;{*0}pi;{-})=(2.78+/-0.24+/-0.25)x10;{-3} and B(B;{0}-->D_{2};{*-}l;{+}nu_{l})xB(D_{2};{*-}-->D;{(*)0}pi;{-})=(1.77+/-0.26+/-0.11)x10;{-3}. In addition we measure the branching ratio Gamma(D_{2};{*}-->Dpi;{-})/Gamma(D_{2};{*}-->D;{(*)}pi;{-})=0.62+/-0.03+/-0.02.