As the deployment of large language models (LLMs) grows in sensitive domains, ensuring the integrity of their computational provenance becomes a critical challenge, particularly in regulated sectors such as healthcare, where strict requirements are applied in dataset usage. We introduce ZKPROV, a novel cryptographic framework that enables zero-knowledge proofs of LLM provenance. It allows users to verify that a model is trained on a reliable dataset without revealing sensitive information about it or its parameters. Unlike prior approaches that focus on complete verification of the training process (incurring significant computational cost) or depend on trusted execution environments, ZKPROV offers a distinct balance. Our method cryptographically binds a trained model to its authorized training dataset(s) through zero-knowledge proofs while avoiding proof of every training step. By leveraging dataset-signed metadata and compact model parameter commitments, ZKPROV provides sound and privacy-preserving assurances that the result of the LLM is derived from a model trained on the claimed authorized and relevant dataset. Experimental results demonstrate the efficiency and scalability of the ZKPROV in generating this proof and verifying it, achieving a practical solution for real-world deployments. We also provide formal security guarantees, proving that our approach preserves dataset confidentiality while ensuring trustworthy dataset provenance.