Numerous soluble proteins convert to insoluble amyloid-like fibrils that have common properties. Amyloid fibrils are associated with fatal diseases such as Alzheimer's, and amyloid-like fibrils can be formed in vitro. For the yeast protein Sup35, conversion to amyloid-like fibrils is associated with a transmissible infection akin to that caused by mammalian prions. A seven-residue peptide segment from Sup35 forms amyloid-like fibrils and closely related microcrystals, from which we have determined the atomic structure of the cross-beta spine. It is a double beta-sheet, with each sheet formed from parallel segments stacked in register. Side chains protruding from the two sheets form a dry, tightly self-complementing steric zipper, bonding the sheets. Within each sheet, every segment is bound to its two neighbouring segments through stacks of both backbone and side-chain hydrogen bonds. The structure illuminates the stability of amyloid fibrils, their self-seeding characteristic and their tendency to form polymorphic structures.