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Using 180 pb-1 of data collected with the CDF II detector at the Tevatron, we measure the first two moments of the hadronic invariant mass-squared distribution in charmed semileptonic B decays. From these we determine the nonperturbative Heavy Quark Effective Theory parameters Λ and λ1 used to relate the B meson semileptonic branching ratio to the CKM matrix element |Vcb|. For a minimum lepton momentum of 0.7 GeV/c in the B rest frame we measure the first two moments of the D**→D(*) π component to be D**2>=(5.83±0.16stat±0.08syst) GeV2/c4 and <(mD**2-D**2>)2>=(1.30±0.69stat±0.22syst) GeV4/c8. Combining these with the discrete mass terms from the D and D* mesons, we find the total moments to be xc2>- m̅D2 =(0.467±0.038stat±0.068syst) GeV2/c4 and <(Mxc2-xc2>)2>=(1.05±0.26stat±0.13syst) GeV4/c8, where m̅D is the spin-averaged D mass. The systematic error is dominated by the uncertainties in the world-average branching ratios used to combine the D, D*, and D**contributions. The analysis makes no assumptions about the shape or resonant structure of the D**→D(*) π invariant mass distribution.