Faithful chromosome segregation and genome maintenance requires the removal of all DNA bridges that physically link chromosomes before cells divide. Using C. elegans embryos we show that the LEM-3/Ankle1 nuclease defines a previously undescribed genome integrity mechanism by processing DNA bridges right before cells divide. LEM-3 acts at the midbody, the structure where abscission occurs at the end of cytokinesis. LEM-3 localization depends on factors needed for midbody assembly, and LEM-3 accumulation is increased and prolonged when chromatin bridges are trapped at the cleavage plane. LEM-3 locally processes chromatin bridges that arise from incomplete DNA replication, unresolved recombination intermediates, or the perturbance of chromosome structure. Proper LEM-3 midbody localization and function is regulated by AIR-2/Aurora B kinase. Strikingly, LEM-3 acts cooperatively with the BRC-1/BRCA1 homologous recombination factor to promote genome integrity. These findings provide a molecular basis for the suspected role of the LEM-3 orthologue Ankle1 in human breast cancer.