Based on the full BABAR data sample, we report improved measurements of the ratios R(D(*))=B(B̅ →D(*)τ-ν̅ τ)/B(B̅ →D(*)ℓℓ-ν̅ ℓ), where ℓ is either e or μ. 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 2.0σ and 2.7σ, respectively. Taken together, our results disagree with these expectations at the 3.4σ level. This excess cannot be explained by a charged Higgs boson in the type II two-Higgs-doublet model.