Fine-grained location prediction on smart phones can be used to improve app/system performance. Application scenarios include video quality adaptation as a function of the 5G network quality at predicted user locations, and augmented reality apps that speed up content rendering based on predicted user locations. Such use cases require prediction error in the same range as the GPS error, and no existing works on location prediction can achieve this level of accuracy. We propose Federated Meta-Location Learning (FMLL) on smart phones for fine-grained location prediction, based on GPSt races collected on the phones. FMLL has three components: a meta-location generation module, a prediction model, and a federated learning framework. The meta-location generation module represents the user location data as relative points in an abstract 2D space, which enables learning across different physical spaces. The model fuses Bidirectional Long Short-Term Memory (BiLSTM) and Convolutional Neural Networks (CNN), where BiLSTM learns the speed and direction of the mobile users, and CNN learns information such as user movement preferences. The framework runs on the phones of the users and also on a server that coordinates learning from all users in the system. FMLL uses federated learning to protect user privacy and reduce bandwidth consumption. Our experimental results, using a dataset with over 600,000 users, demonstrate that FMLL outperforms baseline models in terms of prediction accuracy. We also demonstrate that FMLL works well in conjunction with transfer learning, which enables model reusability. Finally, benchmark results on Android phones demonstrate FMLL's feasibility in real life.