Large language models (LLM) have shown great promise in supporting differential diagnosis, but 23 available published studies on the diagnostic accuracy evaluated small cohorts (number of cases, 30-422, mean 104) and have evaluated LLM responses subjectively by manual curation (23/23 studies). The performance of LLMs for rare disease diagnosis has not been evaluated systematically. Here, we perform a rigorous and large-scale analysis of the performance of a GPT-4 in prioritizing candidate diagnoses, using the largest-ever cohort of rare disease patients. Our computational study used 5267 computational case reports from previously published data. Each case was formatted as a Global Alliance for Genomics and Health (GA4GH) phenopacket, in which clinical anomalies were represented as Human Phenotype Ontology (HPO) terms. We developed software to generate prompts from each phenopacket. Prompts were sent to Generative Pre-trained Transformer 4 (GPT-4), and the rank of the correct diagnosis, if present in the response, was recorded. The mean reciprocal rank of the correct diagnosis was 0.24 (with the reciprocal of the MRR corresponding to a rank of 4.2), and the correct diagnosis was placed in rank 1 in 19.2% of the cases, in the first 3 ranks in 28.6%, and in the first 10 ranks in 32.5%. Our study is the largest to be reported to date and provides a realistic estimate of the performance of GPT-4 in rare disease medicine.