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We introduce λ-Tune, a framework that leverages Large Language Models (LLMs) for automated database system tuning. The design of λ-Tune is motivated by the capabilities of the latest generation of LLMs. Different from prior work, leveraging LLMs to extract tuning hints for single parameters, λ-Tune generates entire configuration scripts, based on a large input document, describing the tuning context. λ-Tune generates alternative configurations, using a principled approach to identify the best configuration, out of a small set of candidates. In doing so, it minimizes reconfiguration overheads and ensures that evaluation costs are bounded as a function of the optimal run time. By treating prompt generation as a cost-based optimization problem, λ-Tune conveys the most relevant context to the LLM while bounding the number of input tokens and, therefore, monetary fees for LLM invocations. We compare λ-Tune to various baselines, using multiple benchmarks and PostgreSQL and MySQL as target systems for tuning, showing that λ-Tune is significantly more robust than prior approaches.