Abstract: An all-to-all coupled, high-fidelity, error-correctable quantum computer can scale to hundreds of qubits within a single cavity of moderate cooperativity with existing neutral atom technology. This quantum processor can enact teleported gates among any pair of qubits using a local Rydberg interaction between each qubit and a separate network of atoms that distribute entanglement via a cavity mode. Small atomic ensembles at network nodes allow for ultrafast, nondestructive readout with high fidelity by substantially shifting the resonance of even a poor-quality cavity. Fast readout enables near-deterministic entanglement distribution among network atoms despite cavity losses as well as syndrome measurements of qubit atoms for error correction.