We introduce FabricDiffuion, a method for transferring fabric textures from a single clothing image to 3D garments of arbitrary shapes. Existing approaches typically synthesize textures on the garment surface through 2D-to-3D texture mapping or depth-aware inpainting via generative models. Unfortunately, these methods often struggles at capturing and preserving texture details, particularly due to challenging occlusions, distortions, or poses in the input image. Inspired by the observation that in fashion industry most garments are constructed by stitching sewing patterns with flat, repeatable textures, we cast the task of clothing texture transfer as extracting distortion-free, tileable texture materials that are subsequently mapped onto the UV space of the garment. Building upon this insight, we train a denoising diffusion model with a large-scale synthetic dataset to rectify distortions in the input texture image. This process yields a flat texture map that enables a tight coupling with existing Physically-Based Rendering (PBR) material generation pipelines, allowing for realistic relighting of the garment under various lighting conditions. We show that FabricDiffusion can transfer a variety of features from a single clothing image including texture patterns, material properties, and detailed prints and logos. Extensive experiments demonstrate that our model significantly outperforms state-to-the-art methods on both synthetic data and real-world, in-the-wild clothing images while generalizing to unseen textures and garment shapes.