Membrane budding-fission is a fundamental process generating intracellular carriers of proteins. Earlier works were focused only on formation of coated buds connected to the initial membrane by narrow membrane necks. We present the theoretical analysis of the whole pathway of budding-fission, including the crucial stage where the membrane neck undergoes fission and the carrier separates from the donor membrane. We consider two successive intermediates of the reaction: 1), a constricted membrane neck coming out of aperture of the assembling protein coat, and 2), hemifission intermediate resulting from self-fusion of the inner monolayer of the neck, while its outer monolayer remains continuous. Transformation of the constricted neck into the hemifission intermediate is driven by the membrane stress produced in the neck by the protein coat. Although apparently similar to hemifusion, the fission is predicted to have an opposite dependence on the monolayer spontaneous curvature. Analysis of the further stages of the process demonstrates that in all practically important cases the hemifission intermediate decays spontaneously into two separate membranes, thereby completing the fission process. We formulate the "job description" for fission proteins by calculating the energy they have to deliver and the radii of the protein coat aperture which have to be reached to drive the fission process.