Careful robot manipulation in every-day cluttered environments requires an accurate understanding of the 3D scene, in order to grasp and place objects stably and reliably and to avoid mistakenly colliding with other objects. In general, we must construct such a 3D interpretation of a complex scene based on limited input, such as a single RGB-D image. We describe SceneComplete, a system for constructing a complete, segmented, 3D model of a scene from a single view. It provides a novel pipeline for composing general-purpose pretrained perception modules (vision-language, segmentation, image-inpainting, image-to-3D, and pose-estimation) to obtain high-accuracy results. We demonstrate its accuracy and effectiveness with respect to ground-truth models in a large benchmark dataset and show that its accurate whole-object reconstruction enables robust grasp proposal generation, including for a dexterous hand.