Amorphous solids are made mainly by cooling the liquid below the glass transition without crystallizing it, a method used since before recorded history1, and by depositing the vapour onto a cold plate2, as well as by several other methods3,4. We report here a new way-by `melting' a solid by pressure below the glass transition of the liquid-and apply it to making a new kind of amorphous ice. Thus, ice I has been transformed to an amorphous phase, as determined by X-ray diffraction, by pressurizing it at 77 K to its extrapolated melting point of 10 kbar. At the melting point, the fluid is well below its glass transition. On heating at a rate of ~2.6 K min-1 at zero pressure it transforms at ~117 K to a second amorphous phase with a heat evolution of 42+/-~8 J g-1 and at ~152 K further transforms to ice I with a heat evolution of 92+/-~15 J g-1. In one sample, ice Ic was formed and in another, existing crystals of ice Ih grew from the amorphous phase. Heating below the 117 K transition causes irreversible changes in the diffraction pattern, and a continuous range of amorphous phases can be made. Similar transformations will probably occur in all solids whose melting point decreases with increasing pressure if they can be cooled sufficiently for a transformation to a crystalline solid to be too slow.