Membrane fusion proceeds via formation of intermediate nonbilayer structures. The stalk model of fusion intermediate is commonly recognized to account for the major phenomenology of the fusion process. However, in its current form, the stalk model poses a challenge. On one hand, it is able to describe qualitatively the modulation of the fusion reaction by the lipid composition of the membranes. On the other, it predicts very large values of the stalk energy, so that the related energy barrier for fusion cannot be overcome by membranes within a biologically reasonable span of time. We suggest a new structure for the fusion stalk, which resolves the energy crisis of the model. Our approach is based on a combined deformation of the stalk membrane including bending of the membrane surface and tilt of the hydrocarbon chains of lipid molecules. We demonstrate that the energy of the fusion stalk is a few times smaller than those predicted previously and the stalks are feasible in real systems. We account quantitatively for the experimental results on dependence of the fusion reaction on the lipid composition of different membrane monolayers. We analyze the dependence of the stalk energy on the distance between the fusing membranes and provide the experimentally testable predictions for the structural features of the stalk intermediates.