  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.
