Human hands play a central role in interacting with other people and objects. For realistic replication of such hand motions, high-fidelity hand meshes have to be reconstructed. In this study, we firstly propose DeepHandMesh, a weakly-supervised deep encoder-decoder framework for high-fidelity hand mesh modeling. We design our system to be trained in an end-to-end and weakly-supervised manner; therefore, it does not require groundtruth meshes. Instead, it relies on weaker supervisions such as 3D joint coordinates and multi-view depth maps, which are easier to get than groundtruth meshes and do not dependent on the mesh topology. Although the proposed DeepHandMesh is trained in a weakly-supervised way, it provides significantly more realistic hand mesh than previous fully-supervised hand models. Our newly introduced penetration avoidance loss further improves results by replicating physical interaction between hand parts. Finally, we demonstrate that our system can also be applied successfully to the 3D hand mesh estimation from general images. Our hand model, dataset, and codes are publicly available.