We present a novel approach to robotic grasp planning using both a learned grasp proposal network and a learned 3D shape reconstruction network. Our system generates 6-DOF grasps from a single RGB-D image of the target object, which is provided as input to both networks. By using the geometric reconstruction to refine the the candidate grasp produced by the grasp proposal network, our system is able to accurately grasp both known and unknown objects, even when the grasp location on the object is not visible in the input image. This paper presents the network architectures, training procedures, and grasp refinement method that comprise our system. Experiments demonstrate the efficacy of our system at grasping both known and unknown objects (91% success rate in a physical robot environment, 84% success rate in a simulated environment). We additionally perform ablation studies that show the benefits of combining a learned grasp proposal with geometric reconstruction for grasping, and also show that our system outperforms several baselines in a grasping task.