Abundant nonfibrillar oligomeric intermediates are a common feature of amyloid formation, and these oligomers, rather than the final fibers, have been suggested to be the toxic species in some amyloid diseases. Whether such oligomers are critical intermediates for fiber assembly or form in an alternate, potentially separable pathway, however, remains unclear. Here we study the polymerization of the amyloidogenic yeast prion protein Sup35. Rapid polymerization occurs in the absence of observable intermediates, and both targeted kinetic and direct single-molecule fluorescence measurements indicate that fibers grow by monomer addition. A three-step model (nucleation, monomer addition, and fiber fragmentation) accurately accounts for the distinctive kinetic features of amyloid formation, including weak concentration dependence, acceleration by agitation, and sigmoidal shape of the polymerization time course. Thus, amyloid growth can occur by monomer addition in a reaction distinct from and competitive with formation of potentially toxic oligomeric intermediates.