The rising pressure on DRAM availability and contract pricing reflects generative AI's massive high-performance memory requirements. This pressure is heavily compounded by hyperscale data center expansion, which now consumes a significant portion of global DRAM output. In this work, we propose a new architecture: Fiber Memory, which reimagines the role of optical fiber in a hyperscale data center, deploying it as an active, recirculating delay-line memory for immutable data, such as large language model weights. We present a data-parallel optical broadcast delay-line memory architecture that accounts for fiber's physical realities. By incorporating space-division multiplexed multi-core fibers, passive optical tap-and-amplify interfaces, co-packaged optics, and regional all-optical regeneration, our case study evaluation suggests that Fiber Memory can eliminate redundant weight storage across 10,000 AI accelerators and reduce weight-delivery energy by over 70% compared to traditional HBM3e configurations.