There are many approaches to weakly-supervised training of networks to segment 2D images. By contrast, existing approaches to segmenting volumetric images rely on full-supervision of a subset of 2D slices of the 3D volume. We propose an approach to volume segmentation that is truly weakly-supervised in the sense that we only need to provide a sparse set of 3D points on the surface of target objects instead of detailed 2D masks. We use the 3D points to deform a 3D template so that it roughly matches the target object outlines and we introduce an architecture that exploits the supervision it provides to train a network to find accurate boundaries. We evaluate our approach on Computed Tomography (CT), Magnetic Resonance Imagery (MRI) and Electron Microscopy (EM) image datasets and show that it substantially reduces the required amount of effort.