We address the problem of 3D shape completion from sparse and noisy point clouds, a fundamental problem in computer vision and robotics. Recent approaches are either data-driven or learning-based: Datadriven approaches rely on a shape model whose parameters are optimized to fit the observations; Learningbased approaches, in contrast, avoid the expensive optimization step by learning to directly predict complete shapes from incomplete observations in a fullysupervised setting. However, full supervision is often not available in practice. In this work, we propose a weakly-supervised learning-based approach to 3D shape completion which neither requires slow optimization nor direct supervision. While we also learn a shape prior on synthetic data, we amortize, i.e., learn, maximum likelihood fitting using deep neural networks resulting in efficient shape completion without sacrificing accuracy. On synthetic benchmarks based on ShapeNet (Chang et al, 2015) and ModelNet (Wu et al, 2015) as well as on real robotics data from KITTI (Geiger et al, 2012) and Kinect (Yang et al, 2018), we demonstrate that the proposed amortized maximum likelihood approach is able to compete with the fully supervised baseline of Dai et al (2017) and outperforms the data-driven approach of Engelmann et al (2016), while requiring less supervision and being significantly faster.