A high-strength hydraulic cement composition contains cement, fine aggregates, coarse aggregates, water and a specified admixture and has a unit amount of water in the range of 120-185 kg/m3. Water and the cement are contained at a ratio of 20-40% and the admixture is contained in the amount of 0.1-3.0 weight parts per 100 weight parts of the cement. The admixture is one or more selected from reaction mixtures obtained by specified two processes and partially or completely neutralized substances obtained from such a reaction mixture through these two processes and also a specified third process. The specified two processes include initiating a radical copolymerization process of a radical polymerizable monomer mixture containing maleic anhydride and a monomer of a specified kind in a non-aqueous system in the presence of a non-aqueous radical initiator and stopping the radical copolymerization reaction in its midst while the inversion rate of polymerization is at a specified level by adding water to the reaction system to thereby obtain a reaction mixture. The specified third process is for partially or completely neutralizing the reaction mixture.