We present measurements of the semileptonic decays B- --> D0 tau- nubar_tau, B- --> D*0 tau- nubar_tau, B0bar --> D+ tau- nubar_tau, and B0bar --> D*+ tau- nubar_tau, which are sensitive to non--Standard Model amplitudes in certain scenarios. The data sample consists of 232x10^6 Upsilon(4S) --> BBbar 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 (ell=e or mu) 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 B --> D(*) tau- nubar_tau (tau- --> ell- nubar_ell nu_tau) events from the backgrounds, predominantly B --> D(*) ell- nubar_ell. We measure the branching-fraction ratios R(D) == BR(B --> D tau- nubar_tau) / BR(B --> D ell- nubar_ell) and R(D*) == BR(B --> D* tau- nubar_tau) / BR(B --> D* ell- nubar_ell) and, from a combined fit to B- and B0bar 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 ell- nubar_ell branching fractions, we obtain BR(B --> D tau- nubar_tau)=(0.86 +/- 0.24 +/- 0.11 +/- 0.06)% and BR(B --> D* tau- nubar_tau)=(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*_ell, and the squared momentum transfer, q^2.