Our ab initio study explains recent puzzling experiments on the conductance of a single Co adatom deposited either on Cu(111) or on ferromagnetic Co islands and contacted with both magnetic and nonmagnetic electrodes [N. Neél, J. Kröger, and R. Berndt, Phys. Rev. Lett. 102, 086805 (2009)]. We provide clear evidence that the conductance of a single atomic junction in the contact regime is close to G₀/2 (G₀ is the quantum of conductance) for ferromagnetic electrodes and to G₀ for nonmagnetic ones. Spin-dependent calculations reveal that a conductance of G₀/2 originates from a combination of partially open majority and minority chanels. The bonding between the Co adatom and the Co island reduces significantly the contribution of the minority d electrons to the conductance, leading to the observation of half-integer conductance.