  We compute corrections of relative order v^4 to the rates for the decays of
^1S_0 heavy quarkonium into two photons and into light hadrons and for the
decays of ^3S_1 heavy quarkonium into a lepton pair and into light hadrons. In
particular, we compute the coefficients of the decay operators that have the
same quantum numbers as the heavy quarkonium. We also confirm previous
calculations of the order-v^2 corrections to these rates. We find that the v
expansion converges well for the decays of ^1S_0 heavy quarkonium and for the
decay of ^3S_1 heavy quarkonium into a lepton pair. Large higher-order-in-v
corrections appear in the decay of ^3S_1 heavy quarkonium into light hadrons.
However, we find that the series of coefficients of operators with ^3S_1
quantum numbers, which yields a large correction in order v^2, yields a smaller
correction in order v^4.
