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 ⟨m2D**⟩=(5.83±0.16stat±0.08syst)  GeV2/c4 and ⟨(m2D**−⟨m2D**⟩)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 ⟨M2Xc⟩−m2D=(0.467±0.038stat±0.068syst)  GeV2/c4 and ⟨(M2Xc−⟨M2Xc⟩)2⟩=(1.05±0.26stat±0.13syst)  GeV4/c8, where mD 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.