Most water ice in the universe is in a form which does not occur naturally on Earth and of which only minimal amounts have been made in the laboratory. We have encountered this ''high-density amorphous ice'' in electron diffraction experiments of low-temperature (T < 30 K) vapor-deposited water and have subsequently modeled its structure using molecular dynamics simulations. The characteristic feature of high-density amorphous ice is the presence of ''interstitiaI'' oxygen pair distances between 3 and 4 Angstrom. However, we find that the structure is best described as a collapsed lattice of the more familiar low-density amorphous form. These distortions are frozen in at temperatures below 38 K because, we propose, it requires the breaking of one hydrogen bond, on average, per molecule to relieve the strain and to restructure the lattice to that of low-density amorphous ice. Several features of astrophysical ice analogs studied in laboratory experiments are readily explained by the structural transition from high-density amorphous ice into low-density amorphous ice. Changes in the shape of the 3.07 mu m water band, trapping efficiency of CO, CO loss, changes in the CO band structure, and the recombination of radicals induced by low-temperature UV photolysis all covary with structural changes that occur in the ice during this amorphous to amorphous transition. While the 3.07 mu m ice band in various astronomical environments can be modeled with spectra of simple mixtures of amorphous and crystalline forms, the contribution of the high-density amorphous form nearly always dominates.