The experimental results on the ratios of branching fractions R(D)=B(B→Dτ-ντ)/B(B→Dℓ-νℓ) and R(D∗)=B(B→D∗τ-ντ)/B(B→D∗ℓ-νℓ), where ℓ denotes an electron or a muon, show a long-standing discrepancy with the standard model predictions, and might hint at a violation of lepton flavor universality. We report a new simultaneous measurement of R(D) and R(D∗), based on a data sample containing 772×106 BB events recorded at the ϒ(4S) resonance with the Belle detector at the KEKB e+e- collider. In this analysis the tag-side B meson is reconstructed in a semileptonic decay mode and the signal-side τ is reconstructed in a purely leptonic decay. The measured values are R(D)=0.307±0.037±0.016 and R(D∗)=0.283±0.018±0.014, where the first uncertainties are statistical and the second are systematic. These results are in agreement with the standard model predictions within 0.2, 1.1, and 0.8 standard deviations for R(D), R(D∗), and their combination, respectively. This work constitutes the most precise measurements of R(D) and R(D∗) performed to date as well as the first result for R(D) based on a semileptonic tagging method.