Imitation learning of robot policies from few demonstrations is crucial in open-ended applications. We propose a new method, Interaction Warping, for learning SE(3) robotic manipulation policies from a single demonstration. We infer the 3D mesh of each object in the environment using shape warping, a technique for aligning point clouds across object instances. Then, we represent manipulation actions as keypoints on objects, which can be warped with the shape of the object. We show successful one-shot imitation learning on three simulated and real-world object re-arrangement tasks. We also demonstrate the ability of our method to predict object meshes and robot grasps in the wild.