Patient-specific implant (PSI) design is a challenging task and requires a specialist, who will spend a significant amount of time using computer aided design tools for implant creation, since patient-specific skull features have to be accounted for. Automating this process could potentially allow intraoperative PSI availability at a relatively low cost. This work proposes to use a 3D Sparse Convolutional Neural Network (SCNN) to reconstruct complete skulls given defective skulls (i.e., skull shape completion) and extract implants by taking the difference between them. With the help of recently published methods for sparse convolutions, it is now possible to avoid the downsampling of the whole skull volume, which is required for conventional dense 3D CNN applications proposed previously. Thus, the SCNN-based approach allows to preserve the original skull geometry. The proposed pipeline includes a supervised SCNN autoencoder network with data preprocessing steps, which further exploit the sparse nature of a skull scan. The best setup in our experiments achieves a Dice Score (DS) of 85.4%, a Border Dice Score of 94.6%, Hausdorff Distance (HD) of 4.91 and 95th percentile HD of 2.64 on the dataset for Task 3 of the AutoImplant 2021 challenge (https://autoimplant2021.grand-challenge.org/ ). The results are comparable with a dense CNN counterpart which has significantly more parameters and requires downsampling and cropping of the skull image on 6GB GPUs. The code is publicly available at https://github.com/akroviakov/SparseSkullCompletion.