The [URE3] phenotype in Saccharomyces cerevisiae propagates by a prion mechanism, involving the aggregation of the normally soluble and highly helical protein Ure2. Previous data have shown that the protein spontaneously forms in vitro long, straight, insoluble fibrils at neutral pH that are similar to amyloids in that they bind Congo red and show green-yellow birefringence and have an increased resistance to proteolysis. These fibrils are not amyloids as they are devoid of a cross-beta core. Here we further document the mechanism of assembly of Ure2p into fibrils. The critical concentration for Ure2p assembly is measured, and the minimal size of the nuclei that are the precursors of Ure2p fibrils is determined. Our data indicate that the assembly process is irreversible. As a consequence, the critical concentration is very low. By analyzing the elongation rates of preformed fibrils and combining the results with single-fiber imaging experiments of a variant Ure2p labeled by fluorescent dyes, we reveal the polarity of the fibrils and differences in the elongation rates at their ends. These results bring novel insight in the process of Ure2p assembly into fibrils and the mechanism of propagation of yeast prions.