Learning-based visual relocalizers exhibit leading pose accuracy, but require hours or days of training. Since training needs to happen on each new scene again, long training times make learning-based relocalization impractical for most applications, despite its promise of high accuracy. In this paper we show how such a system can actually achieve the same accuracy in less than 5 minutes. We start from the obvious: a relocalization network can be split in a scene-agnostic feature backbone, and a scene-specific prediction head. Less obvious: using an MLP prediction head allows us to optimize across thousands of view points simultaneously in each single training iteration. This leads to stable and extremely fast convergence. Furthermore, we substitute effective but slow end-to-end training using a robust pose solver with a curriculum over a reprojection loss. Our approach does not require privileged knowledge, such a depth maps or a 3D model, for speedy training. Overall, our approach is up to 300x faster in mapping than state-of-the-art scene coordinate regression, while keeping accuracy on par. Code is available: https://nianticlabs.github.io/ace