We performed a coincidence measurement of two nucleons emitted from the nonmesonic weak decay of lambda(5)He formed via the 6Li(pi+, K+) reaction. The energies of the two nucleons and the pair number distributions in the opening angle between them were measured. In both np and nn pairs, we observed a clean back-to-back correlation coming from the two-body weak reactions of lambda p --> np and lambda n --> nn, respectively. The ratio of the nucleon pair numbers was N(nn)/N(np) = 0.45 +/- 0.11(stat) +/- 0.03(syst) in the kinematic region of cos theta(NN) < -0.8. Since each decay mode was exclusively detected, the measured ratio should be close to the ratio of gamma(lambda p --> np)/gamma(lambda n --> nn). The ratio is consistent with recent theoretical calculations based on the heavy meson and/or direct-quark exchange picture.