We report a measurement of the ratio R(D*) = B((B) over bar (0) -> D*(+)tau(-)(nu) over bar (tau))/B((B) over bar (0) -> D*(+)l(-)(nu) over bar (l))where l denotes an electron or a muon. The results are based on a data sample containing 772 x 10(6) B (B) over bar pairs recorded at the Upsilon(4S) resonance with the Belle detector at the KEKB e(+)e(-) collider. We select a sample of B-0(B) over bar (0) pairs by reconstructing both B mesons in semileptonic decays to D*(-/+)l(+/-). We measure R(D*) = 0.302 +/- 0.030(stat) +/- 0.011(syst), which is within 1.6 sigma of the Standard Model theoretical expectation, where the standard deviation sigma includes systematic uncertainties. We use this measurement to constrain several scenarios of new physics in a model-independent approach.