Phases of CaCO3 stabilized at high pressures and temperatures are the potential agents of the global carbon cycle, transferring oxidized carbon in deep Earth's interiors and thus are of special interest for the Earth sciences. Here, we report finding of the new phase, named disarag, which is dynamically disordered aragonite with freely rotating CO3 groups, similar to that in the CaCO3-V phase with a calcite-like structure. Disarag has a stability field expanding from 3 to 10 GPa and from 1600 to 2000 K. Consideration of twinned structure enlarges this field, decreasing the transition temperature from aragonite to disarag at 100-300 K. At P-T parameters corresponding to the transition from aragonite to disarag, the marked disappearance of the diffraction peaks is observed in in situ experiments. We show that, among known phases of CaCO3, disarag is the best candidate for the explanation of this reconstruction of diffraction pattern. Also, for the first time, using ab initio molecular dynamics technique, we determine equilibrium curves between calcite and its disordered phases CaCO3-IV and CaCO3-V. We show that the transitions of alkaline-earth carbonates CaCO3, SrCO3, and BaCO3 to the disordered states start when the critical angle of librations of the CO3 group about the axis perpendicular to the molecular three-fold axis exceeds 45 degrees. The calcite-like structure of CaCO3 is characterized by more intense librations than the aragonite-like structure of this compound and reaches the critical angle at lower temperatures. As a result, calcite transforms into the disordered state at lower temperatures than aragonite.