Training large-scale artificial intelligence (AI) models demands significant computational power and energy, leading to increased carbon footprint with potential environmental repercussions. This paper delves into the challenges of training AI models across geographically distributed (geo-distributed) data centers, emphasizing the balance between learning performance and carbon footprint. We consider Federated Learning (FL) as a solution, which prioritizes model parameter exchange over raw data, ensuring data privacy and compliance with local regulations. Given the variability in carbon intensity across regions, we propose a new framework called CAFE (short for Carbon-Aware Federated Learning) to optimize training within a fixed carbon footprint budget. Our approach incorporates coreset selection to assess learning performance, employs the Lyapunov drift-plus-penalty framework to address the unpredictability of future carbon intensity, and devises an efficient algorithm to address the combinatorial complexity of the data center selection. Through extensive simulations using real-world carbon intensity data, we demonstrate the efficacy of our algorithm, highlighting its superiority over existing methods in optimizing learning performance while minimizing environmental impact.