We have studied the evolution of the magnetic properties of monolayer-high cobalt islands on a Pt(111) surface as function of their exposure to oxygen. We observe a sequential quenching of magnetic anisotropy and magnetic moment. For minute exposures to oxygen this leads to an enhancement of the maximum susceptibility of up to 66% for chi". We show that the reason for the reduction of the anisotropy before the one of the moment is due to preferential oxygen adsorption at the island edges, which are the main source of magnetic anisotropy. Our example shows how the enhanced chemical reactivity, combined with the enhanced anisotropy of low coordinated atoms, leads to surprising changes of magnetic properties upon exposure to oxygen.