Large Language Models (LLMs) are increasingly deployed in sensitive domains including healthcare, legal services, and confidential communications, where privacy is paramount. This paper introduces Whisper Leak, a side-channel attack that infers user prompt topics from encrypted LLM traffic by analyzing packet size and timing patterns in streaming responses. Despite TLS encryption protecting content, these metadata patterns leak sufficient information to enable topic classification. We demonstrate the attack across 28 popular LLMs from major providers, achieving near-perfect classification (often >98% AUPRC) and high precision even at extreme class imbalance (10,000:1 noise-to-target ratio). For many models, we achieve 100% precision in identifying sensitive topics like "money laundering" while recovering 5-20% of target conversations. This industry-wide vulnerability poses significant risks for users under network surveillance by ISPs, governments, or local adversaries. We evaluate three mitigation strategies - random padding, token batching, and packet injection - finding that while each reduces attack effectiveness, none provides complete protection. Through responsible disclosure, we have collaborated with providers to implement initial countermeasures. Our findings underscore the need for LLM providers to address metadata leakage as AI systems handle increasingly sensitive information.