In federated learning (FL), data does not leave personal devices when they are jointly training a machine learning model. Instead, these devices share gradients, parameters, or other model updates, with a central party (e.g., a company) coordinating the training. Because data never "leaves" personal devices, FL is often presented as privacy-preserving. Yet, recently it was shown that this protection is but a thin facade, as even a passive, honest-but-curious attacker observing gradients can reconstruct data of individual users contributing to the protocol. In this work, we show a novel data reconstruction attack which allows an active and dishonest central party to efficiently extract user data from the received gradients. While prior work on data reconstruction in FL relies on solving computationally expensive optimization problems or on making easily detectable modifications to the shared model's architecture or parameters, in our attack the central party makes inconspicuous changes to the shared model's weights before sending them out to the users. We call the modified weights of our attack trap weights. Our active attacker is able to recover user data perfectly, i.e., with zero error, even when this data stems from the same class. Recovery comes with near-zero costs: the attack requires no complex optimization objectives. Instead, our attacker exploits inherent data leakage from model gradients and simply amplifies this effect by maliciously altering the weights of the shared model through the trap weights. These specificities enable our attack to scale to fully-connected and convolutional deep neural networks trained with large mini-batches of data. For example, for the high-dimensional vision dataset ImageNet, we perfectly reconstruct more than 50% of the training data points from mini-batches as large as 100 data points.