  We calculate the long-distance matrix elements for the decays of the
lowest-lying S- and P-wave states of charmonium and bottomonium in quenched
lattice QCD, using a nonrelativistic formulation for the heavy quarks. (The
short-distance coefficients are known from perturbation theory.) In particular,
we present the first calculation from QCD first principles of the color-octet
contribution to P-wave decay---a contribution that is absent in potential
models. We also give the relations between the lattice matrix elements and
their continuum counterparts through one-loop order in perturbation theory.
