Disclosed is a sulfur-containing conjugated diolefin copolymer rubber composed of a conjugated diolefin and an aromatic vinyl compound, wherein (1) the content of sulfur bound to the rubber is 0.01 mmol % or more, (2) the content of the aromatic vinyl compound polymerization unit is not less than 5 wt % and less than 45 wt % relative to the amount of the copolymer rubber, and (3) the content of a vinyl bond (a 1,2-bond) in the conjugated diolefin polymerization unit is 30 mol % or more relative to the amount of the conjugated diolefin polymerization unit. In the copolymerization between the conjugated diolefin and the aromatic vinyl compound, a sulfur-containing polymerization terminator is used to terminate the reaction of the copolymerization.