The present invention provides a heat conductive silicone rubber composite sheet that is suitable as a heat dissipating member provided between a heat generating electronic component and a heat dissipating component such as a heat dissipating fin, wherein the heat conductive silicone rubber composite sheet has a laminated structure with good electrical insulation and thermal conductivity, as well as excellent strength, flexibility, and particularly superior interlayer adhesion. The laminated structure has an intermediate layer and a pair of outer layers laminated to both surfaces of the intermediate layer, wherein (A) the intermediate layer is a layer of a synthetic resin film that displays heat resistance and electrical insulation, and (B) the outer layers are silicone rubber layers formed by curing a composition including (a) an organopolysiloxane, (b) a curing agent, (c) a heat conductive filler, and (d) a silicon compound-based adhesion imparting agent with at least one functional group selected from the group consisting of epoxy groups, alkoxy groups, vinyl groups, and the group represented by the formula Si--H.