  Using an effective Hamiltonian derived from the QCD multipole expansion, we
calculate the cross sections of gluo-dissociation of heavy quarkonia in the
quark-gluon plasma, by including both the chromo-electric dipole ($E_1$) as
well as the chromo-magnetic dipole ($M_1$) transition mechanisms. While the
former allows to reproduce the results from operator-product-expansion
calculations in the Coulomb approximation, the latter as a novel contribution
is shown to be significant at low energies close to the threshold. Using thus
obtained cross sections, we further carry out a full calculation of the
gluo-dissociation rates for various charmonia and bottomonia within a
non-relativitic in-medium potential model. The $M_1$ contribution turns out to
be most prominent for the $J/\psi$ and accounts for $\sim 10\%-25\%$ of the
total ($E1+M1$) dissociation rate at temperatures close to the transition
temperature.
