Based on the full BABAR data sample, we report improved measurements of the ratios R(D) = B((B) over bar -> D tau(-)(v) over bar (tau))/B((B) over bar -> Dl(-)(v) over bar (l)) and R(D*) = B((B) over bar -> D*tau(-)(v) over bar (tau))/B((B) over bar -> D*l(-)(v) over bar (l)), where l refers to either an electron or muon. These ratios are sensitive to new physics contributions in the form of a charged Higgs boson. We measure R(D) = 0.440 +/- 0.058 +/- 0.042 and R(D*) = 0.332 +/- 0.024 +/- 0.018, which exceed the standard model expectations by 2.0 sigma and 2.7 sigma, respectively. Taken together, the results disagree with these expectations at the 3.4 sigma level. This excess cannot be explained by a charged Higgs boson in the type II two-Higgs-doublet model. Kinematic distributions presented here exclude large portions of the more general type III two-Higgs-doublet model, but there are solutions within this model compatible with the results.