Improvisation training for actors presents unique challenges, particularly in maintaining narrative coherence and managing cognitive load during performances. Previous research on AI in improvisation performance often predates advances in large language models (LLMs) and relies on human intervention. We introduce ImprovMate, which leverages LLMs as GPTs to automate the generation of narrative stimuli and cues, allowing actors to focus on creativity without keeping track of plot or character continuity. Based on insights from professional improvisers, ImprovMate incorporates exercises that mimic live training, such as abrupt story resolution and reactive thinking exercises, while maintaining coherence via reference tables. By balancing randomness and structured guidance, ImprovMate provides a groundbreaking tool for improv training. Our pilot study revealed that actors might embrace AI techniques if the latter mirrors traditional practices, and appreciate the fresh twist introduced by our approach with the AI-generated cues.