The widespread adoption of large language models (LLMs) has created a pressing need for an efficient, secure and private serving infrastructure, which allows researchers to run open source or custom fine-tuned LLMs and ensures users that their data remains private and is not stored without their consent. While high-performance computing (HPC) systems equipped with state-of-the-art GPUs are well-suited for training LLMs, their batch scheduling paradigm is not designed to support real-time serving of AI applications. Cloud systems, on the other hand, are well suited for web services but commonly lack access to the computational power of HPC clusters, especially expensive and scarce high-end GPUs, which are required for optimal inference speed. We propose an architecture with an implementation consisting of a web service that runs on a cloud VM with secure access to a scalable backend running a multitude of LLM models on HPC systems. By offering a web service using our HPC infrastructure to host LLMs, we leverage the trusted environment of local universities and research centers to offer a private and secure alternative to commercial LLM services. Our solution natively integrates with the HPC batch scheduler Slurm, enabling seamless deployment on HPC clusters, and is able to run side by side with regular Slurm workloads, while utilizing gaps in the schedule created by Slurm. In order to ensure the security of the HPC system, we use the SSH ForceCommand directive to construct a robust circuit breaker, which prevents successful attacks on the web-facing server from affecting the cluster. We have successfully deployed our system as a production service, and made the source code available at \url{https://github.com/gwdg/chat-ai}