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 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

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... he total spin for the occurrence of the Kondo effect which deviates from the usual even-odd alternation.