Numerous proteins have the ability to assemble into fibrillar aggregates which are of great interest, because they feature in scores of human diseases and many technological products. In the present work, we analyze the kinetics of protein fibrillation when the process is governed solely by elongation of initially appeared fibrils in the protein solution. We derive exact expressions for the time dependences of the fibrillation degree, the concentration of monomeric protein in the solution, and the average fibril size. Furthermore, we present formulas for the initial fibrillation rate and the half-fibrillation time in terms of experimentally controllable quantities. The results obtained provide a mechanistic insight into the kinetics of protein fibrillation mediated by fibril elongation. We confront theory with experiment and find that it allows a good description of available experimental data for fibrillation of the Alzheimer's disease-associated protein Aβ(1-40) and the yeast prion protein Sup35.