  We show that a composition operator induced by an analytic self-map of the
unit disc in the complex plane is weakly compact on the space BMOA precisely
when the operator is compact on BMOA. As a crucial step we simplify the
compactness criterion due to Smith for composition operators on BMOA and show
that his condition on the Nevanlinna counting function alone characterizes
compactness. In addition, other equivalent compactness criteria are established
for composition operators on both BMOA and its subspace VMOA.
