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We report the achievement of single-qubit gates with sub-part-per-million error rates, in a trapped-ion ⁴³Ca⁺ hyperfine clock qubit. We explore the speed/fidelity trade-off for gate times 4.4 ≤ t_g ≤ 35 μs, and benchmark a minimum error per single-qubit Clifford of 1.5(4) × 10^-7. Gate calibration errors are suppressed to < 10^-8, leaving qubit decoherence (T₂ ≈ 70 s), leakage, and measurement as the dominant error contributions. The ion is held above a microfabricated surface-electrode trap which incorporates a chip-integrated microwave resonator for electronic qubit control; the trap is operated at room temperature without magnetic shielding.