To understand the mechanism of membrane fusion, we have to infer the sequence of structural transformations that occurs during the process. Here, it is shown how one can estimate the lipid composition-dependent free energies of intermediate structures of different geometries. One can then infer which fusion mechanism is the best explanation of observed behavior in different systems by selecting the mechanism that requires the least energy. The treatment involves no adjustable parameters. It includes contributions to the intermediate energy resulting from the presence of hydrophobic interstices within structures formed between apposed bilayers. Results of these calculations show that a modified form of the stalk mechanism proposed by others is a likely fusion mechanism in a wide range of lipid compositions, but a mechanism based on inverted micellar intermediates (IMIs) is not. This should be true even in the vicinity of the lamellar/inverted hexagonal phase transition, where IMI formation would be most facile. Another prediction of the calculations is that traces of apolar lipids (e.g., long-chain alkanes) in membranes should have a substantial influence on fusion rates in general. The same theoretical methods can be used to generate and refine mechanisms for protein-mediated fusion.