Two structural transitions in covalent aluminum hydride AlH3 were characterized at high pressure. A metallic phase stable above 100 GPa is found to have a remarkably simple cubic structure with shortest first-neighbor H-H distances ever measured except in H2 molecule. Although the high-pressure phase is predicted to be superconductive, this was not observed experimentally down to 4 K over the pressure range 120-164 GPa. The results indicate that the superconducting behavior may be more complex than anticipated.