  We consider transport through a single-molecule magnet strongly coupled to
metallic electrodes. We demonstrate that for half-integer spin of the molecule
electron- and spin-tunneling \emph{cooperate} to produce both quantum tunneling
of the magnetic moment and a Kondo effect in the linear conductance. The Kondo
temperature depends sensitively on the ratio of the transverse and easy-axis
anisotropies in a non-monotonic way. The magnetic symmetry of the transverse
anisotropy imposes a selection rule on the total spin for the occurrence of the
Kondo effect which deviates from the usual even-odd alternation.
