The G-phase, a new metastable phase with its potential energy sitting right in the middle of the glass and crystal, was recently discovered in some simulations when the molten metallic liquid was quenched down to room temperature. In comparison with ordinary glass, the G-phase has a more ordered short-range structure but a similarly disordered long-range structure. The question is whether the G-phase can be termed a new type of glass. In this work, G-phase Cu is made in a molecular dynamics simulation using rapid quenching or isothermal annealing. Weak oscillations are found in the long-range atomic structure. The pseudo-fictive temperature is significantly lower than the Kauzmann temperature; fivefold twinning structures are distinguished in the G-phase whose constituent atoms are face-center-cubic or hexagonal-cubic-packed. This evidence suggests that G-phase Cu is not a glass. However, the G-phase is also metastable against crystallization. Therefore, G-phase Cu is neither a glass nor a crystal but belongs to a new mesophase.