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The ability to distinguish between different movie scenes is critical for understanding the storyline of a movie. However, accurately detecting movie scenes is often challenging as it requires the ability to reason over very long movie segments. This is in contrast to most existing video recognition models, which are typically designed for short-range video analysis. This work proposes a State-Space Transformer model that can efficiently capture dependencies in long movie videos for accurate movie scene detection. Our model, dubbed TranS4mer, is built using a novel S4A building block, which combines the strengths of structured state-space sequence (S4) and self-attention (A) layers. Given a sequence of frames divided into movie shots (uninterrupted periods where the camera position does not change), the S4A block first applies self-attention to capture short-range intra-shot dependencies. Afterward, the state-space operation in the S4A block is used to aggregate long-range inter-shot cues. The final TranS4mer model, which can be trained end-to-end, is obtained by stacking the S4A blocks one after the other multiple times. Our proposed TranS4mer outperforms all prior methods in three movie scene detection datasets, including MovieNet, BBC, and OVSD, while also being $2\times$ faster and requiring $3\times$ less GPU memory than standard Transformer models. We will release our code and models.