ZIF-4, a metal-organic framework (MOF) with a zeolitic structure, undergoes a crystal-amorphous transition on heating to 300 degrees C. The amorphous form, which we term a-ZIF, is recoverable to ambient conditions or may be converted to a dense crystalline phase of the same composition by heating to 400 degrees C. Neutron and x-ray total scattering data collected during the amorphization process are used as a basis for reverse Monte Carlo refinement of an atomistic model of the structure of a-ZIF. The structure is best understood in terms of a continuous random network analogous to that of a-SiO2. Optical microscopy, electron diffraction and nanoindentation measurements reveal a-ZIF to be an isotropic glasslike phase capable of plastic flow on its formation. Our results suggest an avenue for designing broad new families of amorphous and glasslike materials that exploit the chemical and structural diversity of MOFs.