As AI systems become more autonomous and capable, experts warn of them potentially causing catastrophic losses. Drawing on the successful precedent set by the nuclear power industry, this paper argues that developers of frontier AI models should be assigned limited, strict, and exclusive third party liability for harms resulting from Critical AI Occurrences (CAIOs) - events that cause or easily could have caused catastrophic losses. Mandatory insurance for CAIO liability is recommended to overcome developers' judgment-proofness, mitigate winner's curse dynamics, and leverage insurers' quasi-regulatory abilities. Based on theoretical arguments and observations from the analogous nuclear power context, insurers are expected to engage in a mix of causal risk-modeling, monitoring, lobbying for stricter regulation, and providing loss prevention guidance in the context of insuring against heavy-tail risks from AI. While not a substitute for regulation, clear liability assignment and mandatory insurance can help efficiently allocate resources to risk-modeling and safe design, facilitating future regulatory efforts.