A model is presented which proposes a specific cause-and-effect relationship between a limited cell division potential and the maximum lifespan of humans and other mammals. It is based on the clonal succession hypothesis of Kay which states that continually replicating cell beds (e.g. bone marrow, intestinal crypts, epidermis) could be composed of cells with short, well-defined division potentials. In this model, the cells of these beds are proposed to exist in an ordered hierarchy which establishes a specific sequence for cell divisions throughout the organism's lifespan. The depletion of division potential at all hierarchical levels leads to a loss of bed function and sets an intrinsic limit to species longevity. A specific hierarchy for cell proliferation is defined which allows the calculation of time to bed depletion and, ultimately, to organism mortality. The model allows the existence of a small number (n) of critical cell beds within the organism and defines organism death as the inability of any one of these beds to produce cells. The model is consistent with all major observations related to cellular and organismic aging. In particular, it links the PDLs (population doubling limit) observed for various species to their mean lifespan; it explains the slow decline in PDL as a function of age of the donor; it establishes a thermodynamically stable maximum lifespan for a disease-free population; and it can explain why tissue transplants outlive donors or hosts.