We performed a coincidence measurement of two nucleons emitted from the nonmesonic weak decay (NMWD) of ^{5}_{Lambda}He formed via the ^{6}Li(pi^+,K^+) reaction. The energies of 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 decay of Lambda p --> n p and Lambda n --> n n, respectively. The ratio of the nucleon pair numbers was N_{nn}/N_{np}=0.45 pm 0.11(stat)pm 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/direct quark exchange picture.
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