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Single magnetic atoms on surfaces are the smallest conceivable units for two-dimensional magnetic data storage. Previous experiments on such systems have investigated magnetization curves^,, the many-body Kondo effect^, and magnetic excitations in quantum spin systems^,, but a stable magnetization has not yet been detected for an atom on a non-magnetic surface in the absence of a magnetic field. The spin direction of a single magnetic atom can be fixed by coupling it to an underlying magnetic substrate via the exchange interaction^,, but it is then difficult to differentiate between the magnetism of the atom and the surface. Here, we take advantage of the orbital symmetry of the spin-polarized density of states of single cobalt atoms to unambiguously determine their spin direction in real space using a combination of spin-resolved scanning tunnelling microscopy experiments and ab initio calculations. By laterally …