Provided are a one-pack addition curable silicone composition compatible with both a good long-term storage property at room temperature under conditions whereby air is blocked to a constant level and the advance of an addition curing reaction at room temperature by being applied in a thin film of 1500 µm or less and exposed to air, which could not be obtained by conventional curable heat-dissipating grease, and a method for storing the same, and a method for curing this composition. A one-pack addition curable silicone composition having as essential ingredients: (A) organopolysiloxane having silicon-atom-bonded aliphatic unsaturated hydrocarbon groups and a specific kinematic viscosity; (B) one or more thermally conductive fillers selected from metals, metal oxides, metal hydroxides, metal nitrides, metal carbides, and allotropes of carbon; (C) organohydrogenpolysiloxane; and (D) a platinum group metal complex having an organic phosphorus compound represented by formula (1)
