We report the achievement of single-qubit gates with sub-part-per-million error rates, in a trapped-ion ^{43}Ca^{+} hyperfine clock qubit. We explore the speed and fidelity trade-off for gate times 4.4≤t_{g}≤35 μs, and benchmark a minimum error per Clifford gate of 1.5(4)×10^{-7}. Calibration errors are suppressed to <10^{-8}, leaving qubit decoherence (T_{2}≈70 s), leakage, and measurement as the dominant error contributions. The ion is held above a microfabricated surface-electrode trap that incorporates a chip-integrated microwave resonator for electronic qubit control; the trap is operated at room temperature without magnetic shielding.