Aggregation of proteins and peptides is a widespread and much-studied problem, with serious implications in contexts ranging from biotechnology to human disease. An understanding of the proliferation of such aggregates under specific conditions requires a quantitative knowledge of the kinetics and thermodynamics of their formation; measurements that to date have remained elusive. Here, we show that precise determination of the growth rates of ordered protein aggregates such as amyloid fibrils can be achieved through real-time monitoring, using a quartz crystal oscillator, of the changes in the numbers of molecules in the fibrils from variations in their masses. We show further that this approach allows the effect of other molecular species on fibril growth to be characterized quantitatively. This method is widely applicable, and we illustrate its power by exploring the free-energy landscape associated with the conversion of the protein insulin to its amyloid form and elucidate the role of a chemical chaperone and a small heat shock protein in inhibiting the aggregation reaction.