In the last decades, it has been demonstrated that many animal species orient in the Earth magnetic field. One of the best-studied examples is the use of the geomagnetic field by migratory birds for orientation and navigation. However, the biophysical mechanism underlying animal magnetoreception is still not understood. One theory for magnetoreception in birds invokes the so-called radical-pair model. This mechanism involves a pair of reactive radicals, whose chemical fate can be influenced by the orientation with respect to the magnetic field of the Earth through Zeeman and hyperfine interactions. The fact that the geomagnetic field is weak, i.e., ~0.5 G, puts a severe constraint on the radical pair that can establish the magnetic compass sense. For a noticeable change of the reaction yield in a redirected geomagnetic field, the hyperfine interaction has to be as weak as the Earth field Zeeman interaction, i.e., unusually weak for an organic compound. Such weak hyperfine interaction can be achieved if one of the radicals is completely devoid of this interaction as realized in a radical pair containing an oxygen molecule as one of the radicals. Accordingly, we investigate here a possible radical pair-based reaction in the photoreceptor cryptochrome that reduces the protein's flavin group from its signaling state FADH$^bullet$ to the inactive state FADH$^–$ (which reacts to the likewise inactive FAD) by means of the superoxide radical, O2$^$. We argue that the spin dynamics in the suggested reaction can act as a geomagnetic compass and that the very low physiological concentration (nM-$M) of otherwise toxic $^$ is sufficient, even favorable, for the biological function.