While generative artificial intelligence (AI) has shown promise for biological design, no computational system has yet created and validated sequences capable of autonomous replication. Focusing on plasmids, one of the smallest types of replicating elements, we develop OriGen, a sequence model that generates new plasmid origins of replication while maintaining key elements essential for their replication. We experimentally validate OriGen's ability to create functional origins that diverge from existing wild types, demonstrating the model's capacity to capture the complex and often cryptic mechanisms of biological replication.