Ammonia is an important compound with many uses, such as in the manufacture of fertilizers, explosives and pharmaceuticals. As an archetypal hydrogen-bonded system, the properties of ammonia under pressure are of fundamental interest, and compressed ammonia has a significant role in planetary physics. We predict new high-pressure crystalline phases of ammonia (NH3) through a computational search based on first-principles density-functional-theory calculations(1). Ammonia is known to form hydrogen-bonded solids(2-6), but we predict that at higher pressures it will form ammonium amide ionic solids consisting of alternate layers of NH4+ and NH2-ions. These ionic phases are predicted to be stable over a wide range of pressures readily obtainable in laboratory experiments. The occurrence of ionic phases is rationalized in terms of the relative ease of forming ammonium and amide ions from ammonia molecules, and the volume reduction on doing so. We also predict that the ionic bonding cannot be sustained under extreme compression and that, at pressures beyond the reach of current static-loading experiments, ammonia will return to hydrogen-bonded structures consisting of neutral NH3 molecules.