Abstract: A nitrile group-containing, highly saturated copolymer rubber is disclosed, the copolymer chain of which is comprised of (1) 5 to 40% by weight of units of an unsaturated nitrile monomer, (2) 1 to 80% by weight of units of a monomer selected from a fluorine-free unsaturated carboxylic acid ester monomer and a fluorine-containing vinyl monomer and (3) up to 20% by weight of units of a conjugated diene monomer, with the balance being (4) units of a hydrogenated conjugated diene monomer, wherein the sum of the contents of the monomer units (1) and (2) is 30 to 90% and the sum of the contents of the monomer units (3) and (4) is 10 to 70% by weight. This copolymer rubber results in a rubber composition having an improved cold resistance.