  The complete fusion of $^9$Be with $^{144}$Sm and $^{208}$Pb targets is
calculated in the coupled-channels approach. The calculations include couplings
between the 3/2$^-$, 5/2$^-$, and 7/2$^-$ states in the $K=3/2$ ground state
rotational band of $^9$Be. It is shown that the $B(E2)$ values for the
excitation of these states are accurately described in the rotor model. The
interaction of the strongly deformed $^9$Be nucleus with a spherical target is
calculated using the double-folding technique and the effective M3Y
interaction, which is supplemented with a repulsive term that is adjusted to
optimize the fit to the data for the $^{144}$Sm target. The complete fusion is
described by in-going-wave boundary conditions. The decay of the unbound
excited states in $^9$Be is considered explicitly in the calculations by using
complex excitation energies. The model gives an excellent account of the
complete fusion (CF) data for $^9$Be+$^{144}$Sm, and the cross sections for the
decay of the excited states are in surprisingly good agreement with the
incomplete fusion (ICF) data. Similar calculations for $^9$Be+$^{208}$Pb
explain the total fusion data at high energies but fail to explain the CF data,
which are suppressed by 20%, and the calculated cross section for the decay of
excited states is a factor of three smaller than the ICF data at high energies.
Possible reasons for these discrepancies are discussed.
