Silicone rubber parts having a high thermal conductivity and low hardness are obtained by molding and curing a composition comprising (A) 100 parts by weight of an alkenyl-bearing organopolysiloxane having a viscosity of up to about 500,000 centistokes at 25° C., (B) 300-1,200 parts by weight of aluminum oxide powder, (C) 0.05-10 parts by weight of an alkoxysilane of the formula: R1aSi(OR2)(4−a) wherein R1 is a monovalent C6-20 hydrocarbon group, R2 is a C1-6 alkyl group, and a=1, 2 or 3, (D) a platinum group metal catalyst, and (E) an organohydrogenpolysiloxane containing at least two SiH groups in a molecule. The composition is formulated so as to suppress the viscosity increase of a liquid silicone rubber composition due to heavy loading of aluminum oxide.