Using quantitative measurements of protein aggregation rates, we develop a kinetic picture of protein conversion from a soluble to a fibrillar state which shows that a single free energy barrier to aggregation controls the addition of protein molecules into amyloid fibrils, while the characteristic sublinear concentration dependence emerges as a natural consequence of finite diffusion times. These findings suggest that this reaction does not follow a simple chemical mechanism, but rather operates in a way analogous to the landscape models of protein folding defined by stochastic dynamics on a characteristic energy surface.