A rubber composition for a golf ball core includes (a) a conjugated diene polymer, (b) a copolymer of a conjugated diene compound and a non-conjugated olefin, and (c) an organic peroxide. Golf balls having at least one core layer formed of such a rubber composition and a cover of one or more layers encasing the core exhibit improved durability to cracking on repeated impact while maintaining the basic properties required of golf balls, such as flight and spin controllability on approach shots.