Pruning seeks to design lightweight architectures by removing redundant weights in overparameterized networks. Most of the existing techniques first remove structured subnetworks (filters, channels,...) and then fine-tune the resulting networks to maintain a high accuracy. However, removing a whole structure is a strong topological prior and recovering the accuracy, with fine-tuning, is highly cumbersome. In this paper, we introduce an "end-to-end" lightweight network design that achieves training and pruning simultaneously without fine-tuning. The design principle of our method relies on reparametrization that learns not only the weights but also the topological structure of the lightweight sub-network. This reparametrization acts as a prior (or regularizer) that defines pruning masks implicitly from the weights of the underlying network, without increasing the number of training parameters. Sparsity is induced with a budget loss that provides an accurate pruning. Extensive experiments conducted on the CIFAR10 and the TinyImageNet datasets, using standard architectures (namely Conv4, VGG19 and ResNet18), show compelling results without fine-tuning.