We propose a novel data-driven approach for automatically detecting and completing gaps in line drawings with a Convolutional Neural Network. In the case of existing inpainting approaches for natural images, masks indicating the missing regions are generally required as input. Here, we show that line drawings have enough structures that can be learned by the CNN to allow automatic detection and completion of the gaps without any such input. Thus, our method can find the gaps in line drawings and complete them without user interaction. Furthermore, the completion realistically conserves thickness and curvature of the line segments. All the necessary heuristics for such realistic line completion are learned naturally from a dataset of line drawings, where various patterns of line completion are generated on the fly as training pairs to improve the model generalization. We evaluate our method qualitatively on a diverse set of challenging line drawings and also provide quantitative results with a user study, where it significantly outperforms the state of the art.