Manipulating matter at the nanometre scale is important for many electronic, chemical and biological advances, but present solid-state fabrication methods do not reproducibly achieve dimensional control at the nanometre scale. Here we report a means of fashioning matter at these dimensions that uses low-energy ion beams and reveals surprising atomic transport phenomena that occur in a variety of materials and geometries. The method is implemented in a feedback-controlled sputtering system that provides fine control over ion beam exposure and sample temperature. We call the method "ion-beam sculpting", and apply it to the problem of fabricating a molecular-scale hole, or nanopore, in a thin insulating solid-state membrane. Such pores can serve to localize molecular-scale electrical junctions and switches and function as masks to create other small-scale structures. Nanopores also function as membrane channels in all living systems, where they serve as extremely sensitive electro-mechanical devices that regulate electric potential, ionic flow, and molecular transport across cellular membranes. We show that ion-beam sculpting can be used to fashion an analogous solid-state device: a robust electronic detector consisting of a single nanopore in a Si3N4 membrane, capable of registering single DNA molecules in aqueous solution.