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The propensity of large language models (LLMs) to generate hallucinations and non-factual content undermines their reliability in high-stakes domains, where rigorous control over Type I errors (the conditional probability of incorrectly classifying hallucinations as truthful content) is essential. Despite its importance, formal verification of LLM factuality with such guarantees remains largely unexplored.In this paper, we introduce FactTest, a novel framework that statistically assesses whether an LLM can provide correct answers to given questions with high-probability correctness guarantees. We formulate hallucination detection as a hypothesis testing problem to enforce an upper bound of Type I errors at user-specified significance levels. Notably, we prove that FactTest also ensures strong Type II error control under mild conditions and can be extended to maintain its effectiveness when covariate shifts exist. Our approach is distribution-free and works for any number of human-annotated samples. It is model-agnostic and applies to any black-box or white-box LM. Extensive experiments on question-answering (QA) benchmarks demonstrate that FactTest effectively detects hallucinations and enable LLMs to abstain from answering unknown questions, leading to an over 40% accuracy improvement.