A nitrile-group-containing highly saturated copolymer rubber comprising α,β-ethylenically unsaturated nitrile monomer units (a) in 5 to 60 wt %, α,β-ethylenically unsaturated dicarboxylic acid mono ester monomer units (b) in 0.1 to 20 wt %, (meth)acrylic acid alkoxyalkyl ester monomer units which have alkoxyalkyl groups with 2 to 8 carbon atoms (c) in 11 to 50 wt %, and conjugated diene monomer units (d) in 20 to 83.9 wt %, where at least part of the conjugated diene monomer units (d) are hydrogenated.