Self-perpetuating changes in the conformations of amyloidogenic proteins play vital roles in normal biology and disease. Despite intense research, the architecture and conformational conversion of amyloids remain poorly understood. Amyloid conformers of Sup35 are the molecular embodiment of the yeast prion known as [PSI], which produces heritable changes in phenotype through self-perpetuating changes in protein folding. Here we determine the nature of Sup35's cooperatively folded amyloid core, and use this information to investigate central questions in prion biology. Specific segments of the amyloid core form intermolecular contacts in a 'Head-to-Head', 'Tail-to-Tail' fashion, but the 'Central Core' is sequestered through intramolecular contacts. The Head acquires productive interactions first, and these nucleate assembly. Variations in the length of the amyloid core and the nature of intermolecular interfaces form the structural basis of distinct prion 'strains', which produce variant phenotypes in vivo. These findings resolve several problems in yeast prion biology and have broad implications for other amyloids.