We present measurements of the semileptonic decays B-→D0τ- ν̄τ, B-→D*0τ-ν̄τ, B̄0→D+τ- ν̄τ, and B̄0→D*+τ-ν̄τ, which are sensitive to non-standard model amplitudes in certain scenarios. The data sample consists of 232×106Υ(4S)→BB̄ decays collected with the BABAR detector at the PEP-II e+e- collider. We select events with a D or D* meson and a light lepton (=e or μ) recoiling against a fully reconstructed B meson. We perform a fit to the joint distribution of lepton momentum and missing mass squared to distinguish signal B̄→D(*) τ-ν̄τ(τ-→-ν̄ντ) events from the backgrounds, predominantly B̄→D(*)-ν̄. We measure the branching-fraction ratios R(D)B(B̄→Dτ-ν̄τ)/ B(B̄→D-ν̄) and R(D*)B(B̄→D*τ- ν̄τ)/B(B̄→D*-ν̄) and, from a combined fit to B- and B̄0 channels, obtain the results R(D)=(41.6±11.7±5.2) % and R(D*)=(29.7±5.6±1.8)%, where the uncertainties are statistical and systematic. Normalizing to measured B-→D(*)0- ν̄ branching fractions, we obtain B(B̄→Dτ-ν̄τ) =(0.86±0.24±0.11±0.06)% and B(B̄→D*τ- ν̄τ)=(1.62±0.31±0.10±0.05)%, where the additional third uncertainty is from the normalization mode. We also present, for the first time, distributions of the lepton momentum, |p*|, and the squared momentum transfer, q2. © 2009 The American Physical Society.