We analyze the electronic transport through a model spin-1 molecule as a function of temperature, magnetic field and bias voltage. We consider the effect of magnetic anisotropy, which can be generated experimentally by stretching the molecule. In the experimentally relevant regime the conductance of the unstretched molecule reaches the unitary limit of the underscreened spin-1 Kondo effect at low temperatures. The magnetic anisotropy generates an antiferromagnetic coupling between the remaining spin-1/2 and a singular density of quasiparticles, producing a second Kondo effect and a reduced conductance. The results explain recent measurements in spin-1 molecules [Parks J.J. et al., Science, 328 (2010) 1370].