The semileptonic decay channel B -textgreater D tau nu is sensitive to the presence of a scalar current, such as that mediated by a charged-Higgs boson. Recently the BaBar experiment reported the first observation of the exclusive semileptonic decay B -textgreater D tau nu, finding an approximately 2-sigma disagreement with the Standard-Model prediction for the ratio R(D)=BR(B-textgreaterD tau nu)/BR(B-textgreaterD l nu), where l=e,mu. We compute this ratio of branching fractions using hadronic form factors computed in unquenched lattice QCD and obtain R(D) = 0.316(12)(7), where the errors are statistical and total systematic, respectively. This result is the first Standard-Model calculation of R(D) from ab initio full QCD. Its error is smaller than that of previous estimates, primarily due to the reduced uncertainty in the scalar form factor f_0(qtextasciicircum2). Our determination of R(D) is approximately 1-sigma higher than previous estimates and, thus, reduces the tension with experiment. We also compute R(D) in models with electrically charged scalar exchange, such as the type II two-Higgs doublet model. Once again, our result is consistent with, but approximately 1-sigma higher than, previous estimates for phenomenologically relevant values of the scalar coupling in the type II model. As a byproduct of our calculation, we also present the Standard-Model prediction for the longitudinal polarization ratio P_L (D)= 0.325(4)(3).