  We implement all single-qubit operations with fidelities significantly above
the minimum threshold required for fault-tolerant quantum computing, using a
trapped-ion qubit stored in hyperfine "atomic clock" states of $^{43}$Ca$^+$.
We measure a combined qubit state preparation and single-shot readout fidelity
of 99.93%, a memory coherence time of $T^*_2=50$ seconds, and an average
single-qubit gate fidelity of 99.9999%. These results are achieved in a
room-temperature microfabricated surface trap, without the use of magnetic
field shielding or dynamic decoupling techniques to overcome technical noise.
