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 easyaxis 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.PACS numbers: 72.10. Fk, 75.10.Jm, 75.30.Gw, 75.60.Jk Introduction. Single-molecule magnets (SMMs) such as Mn 12 or Fe 8 have been the focus of intense experimental and theoretical investigation [1]. These molecules are characterized by a large spin (S > 1/2), easy-axis and transverse anisotropies, and weak intermolecular interaction. Molecular-crystal properties are due to an ensemble of single molecules and exhibit quantum tunneling of magnetization (QTM) on a mesoscopic scale. Recently, a single molecule magnet (Mn 12 ) was trapped in a nanogap [2,3] and fingerprints of the molecular spin were observed in electron transport. Furthermore, transport fingerprints of QTM were predicted [4] when the individual excitations can be resolved by the temperature. Using easy-axis anisotropy for magnetic device operation was also proposed [5]. These works focused on the regime where single electrons charge and discharge the molecule through weak tunneling. In this Letter we investigate linear transport through a half-integer spin SMM deep inside the blockade regime [6] where the charge on the molecule remains fixed. A strong tunnel-coupling to the metallic electrodes induces spin fluctuations and allows the magnetic moment to tunnel. This is remarkable, since for an isolated SMM with half-integer S this is forbidden by time-reversal (TR) symmetry. At the same time, the resonant spinscattering allows electrons to pass through the SMM: the Kondo effect for transport [7,8] results in a zero bias conductance anomaly that has been studied experimentally in many systems with small spins (e.g. quantum dots [9,10,11,12,13] and single molecules [14,15]). Such an effect is unexpected in SMMs because the S > 1/2 underscreened Kondo effect is suppressed by the easyaxis anisotropy barrier which freezes the spin along the easy axis. However, we find that even a weak transverse anisotropy induces a pseudo-spin-1/2 Kondo effect. The corresponding Kondo temperature is experimentally accessible due to a compensation by the large value of the physical spin S. We perform a scaling analysis [16] for the effective pseudo-spin-1/2 model and verify the results by a non-perturbative numerical renormalization group (NRG) calculation [17,18] for the full large-spin Hamiltonian.Model. We consider SMMs which can be described by the following minimal model in the limit of strong tunnel-