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The field of quantum communications promises the faithful distribution of quantum information, quantum entanglement, and absolutely secret keys, however, the highest rates of these tasks are fundamentally limited by the transmission distance between quantum repeaters. The ultimate end-to-end rates of quantum communication networks are known to be achievable by an optimal entanglement distillation protocol followed by teleportation. In this work, we give a practical design for this achievability. Our ultimate design is an iterative approach, where each purification step operates on shared entangled states and detects loss errors at the highest rates allowed by physics. As a simpler design, we show that the first round of iteration can purify completely at high rates. We propose an experimental implementation using linear optics and photon-number measurements which is robust to inefficient operations and measurements, showcasing its near-term potential for real-world practical applications.