We present Conformal Intent Classification and Clarification (CICC), a framework for fast and accurate intent classification for task-oriented dialogue systems. The framework turns heuristic uncertainty scores of any intent classifier into a clarification question that is guaranteed to contain the true intent at a pre-defined confidence level.By disambiguating between a small number of likely intents, the user query can be resolved quickly and accurately. Additionally, we propose to augment the framework for out-of-scope detection.In a comparative evaluation using seven intent recognition datasets we find that CICC generates small clarification questions and is capable of out-of-scope detection.CICC can help practitioners and researchers substantially in improving the user experience of dialogue agents with specific clarification questions.