Human populations in many countries have undergone a phase of demographic transition, characterized by a major reduction in fertility at a time of increased resource availability. A key stylized fact is that the reduction in fertility is preceded by a reduction in mortality and a consequent increase in population density. Various theories have been proposed to account for the demographic transition process, including maladaptation, increased parental investment in fewer offspring, and cultural evolution. None of these approaches, including formal cultural evolutionary models of the demographic transitions, have addressed a possible direct causal relationship between a reduction in mortality and the subsequent decline in fertility. We provide mathematical models in which low mortality favours the cultural selection of low-fertility traits. This occurs because reduced mortality slows turnover in the model, which allows the cultural transmission advantage of low-fertility traits to outrace their reproductive disadvantage. For mortality to be a crucial determinant of outcome, a cultural transmission bias is required where slow reproducers exert higher social influence. Computer simulations of our models that allow for exogenous variation in the death rate can reproduce the central features of the demographic transition process, including substantial reductions in fertility within only one to three generations. A model assuming continuous evolution of reproduction rates through imitation errors predicts fertility to fall below replacement levels if death rates are sufficiently low. This can potentially explain the very low preferred family sizes in Western Europe.