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 Lambda and lambda(1) used to relate the B meson semileptonic branching ratio to the CKM matrix element vertical bar V(ch)vertical bar. 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((*))pi component to be < m(D**)(2)> = (5.83 +/- 0.16(stat) +/- 0.08(syst)) GeV2/c(4) and <(M-D**(2) - < m(D**)(2)>)(2)> = (130 +/- 0.69(stat) +/- 0.22(syst)) GeV4/c(8). Combining these with the discrete mass terms from the D and D* mesons, we find the total moments to be < M-Xc(2)> - (m) over bar (2)(D) = (0.467 +/- 0.038(stat) +/- 0.068(syst)) GeV2/c(4) and <(M-Xc(2) - < M-Xc(2)>)(2)) = (1.05 +/- 0.26(stat) +/- 0.13(syst)) GeV4/c(8), where (m) over bar (2)(D) 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((*))pi invariant mass distribution.