When a heavy atomic nucleus splits (fission), the resulting fragments are observed to emerge spinning(1); this phenomenon has been a mystery in nuclear physics for over 40 years(2,3). The internal generation of typically six or seven units of angular momentum in each fragment is particularly puzzling for systems that start with zero, or almost zero, spin. There are currently no experimental observations that enable decisive discrimination between the many competing theories for the mechanism that generates the angular momentum(4-12). Nevertheless, the consensus is that excitation of collective vibrational modes generates the intrinsic spin before the nucleus splits (pre-scission). Here we show that there is no significant correlation between the spins of the fragment partners, which leads us to conclude that angular momentum in fission is actually generated after the nucleus splits (post-scission). We present comprehensive data showing that the average spin is strongly mass-dependent, varying in saw-tooth distributions. We observe no notable dependence of fragment spin on the mass or charge of the partner nucleus, confirming the uncorrelated post-scission nature of the spin mechanism. To explain these observations, we propose that the collective motion of nucleons in the ruptured neck of the fissioning system generates two independent torques, analogous to the snapping of an elastic band. A parameterization based on occupation of angular momentum states according to statistical theory describes the full range of experimental data well. This insight into the role of spin in nuclear fission is not only important for the fundamental understanding and theoretical description of fission, but also has consequences for the gamma-ray heating problem in nuclear reactors(13,14), for the study of the structure of neutron-rich isotopes(15,16), and for the synthesis and stability of super-heavy elements(17,18). gamma-ray spectroscopy experiments on the origin of spin in the products of nuclear fission of spin-zero nuclei suggest that the fission fragments acquire their spin after scission, rather than before.