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Quantum electrodynamics has been tested to accuracies below the parts-per-trillion level in light-mass systems. However, tests in heavy-mass systems with a large nuclear charge have not yet reached similar accuracy. Here we report the hyperfine-structure splitting in the 1s ground state of radioactive hydrogen-like 208Bi82+. We produced the isotope in a nuclear reaction and injected the beam into a storage ring to perform laser spectroscopy on samples of 105 ions of Bi82+ that have only a single remaining electron, which experiences extreme magnetic-field strengths. Our result for the hyperfine splitting is in excellent agreement with the most accurate prediction based on a combination of quantum electrodynamics calculations with an empirical treatment of the hyperfine-structure anomaly ratio extracted from laser spectroscopy on neutral atoms of 209Bi and 208Bi. This achievement paves the way for the most stringent test of quantum electrodynamics in strong magnetic fields and demonstrates the feasibility of laser spectroscopy on other exotic ions with low production yields.