The stability of tearing modes is analysed for current density profiles with a small, well localized, additional current around the resonant mode rational surface. The stability parameter Δ' is calculated with a perturbative treatment of the additional current. For the case of a Gaussian profile of the additional current, this is shown to lead to a simple and illustrative condition for the amount of additional current required for stabilization. The consequences of this condition are analysed. In particular, the dependence of the effect of the additional current on its exact location is clearly shown. In the case of a co-driven additional current, far less current is required than in the case of a counter-driven current.