We have measured the magnetic and thermal properties below 2 K of a quasi-1-D frustrated spin-1/2 system Cs2Cu2Mo3O12 called CuO2 ribbon chain systems in order to investigate its magnetic ground state. In the temperature dependence of the specific heat divided by temperature, C/T-T, we found a sharp peak structure at T-N = 1.85 K, which indicates the existence of a phase transition. Considering the behavior of the magnetization and specific heat, it is found that Cs2Cu2Mo3O12 exhibits an antiferromagnetic transition at TN = 1.85 K, and its magnetic ground state is an antiferromagnetic long-range ordered one. The determined ground state of Cs2Cu2Mo3O12 significantly contrasts with a nonmagnetic ground state of isostructural compound Rb2Cu2Mo3O12. We discuss the origins of difference of the magnetic ground states between Rb2Cu2Mo3O12 and Cs2Cu2Mo3O12.