THE discovery1 in 1984 that an ice crystal can be transformed by pressure to an amorphous phase has since been followed by other examples of pressure-induced amorphization2. This transition, like melting, involves loss of long-ranged order, prompting the question of whether the two transitions are related. Here I describe experiments probing this relationship for a form of crystalline ice (denoted Ih) which is melted and amorphized by pressure. To avoid the complication of crystal-crystal transformations interrupting the melting process I use an ice emulsion, in which the very small particle size (about 5μm) suppresses nucleation of other crystal phases. As the temperature is decreased, I see a smooth crossover from (pressure-induced) equilibrium melting to sluggish amorphization at around 140-165 K. In this temperature range, ice Ih became 'supercompressed' before melting to a highly viscous liquid which seemed to be related to an imperfectly relaxed amorphous ice. Below about 140 K, ice Ih was transformed to an unrelaxed phase apparently related to the high-density amorphous form of ice. This sequence of transitions can be viewed as a crossover from a two-phase melting process (which is determined by the relative free energies of the solid and liquid phases) towards a one-phase amorphization process (where the transition is induced by a mechanical instability limit of the solid).