  We demonstrate single-shot qubit readout with fidelity sufficient for
fault-tolerant quantum computation, for two types of qubit stored in single
trapped calcium ions. For an optical qubit stored in the (4S_1/2, 3D_5/2)
levels of 40Ca+ we achieve 99.991(1)% average readout fidelity in one million
trials, using time-resolved photon counting. An adaptive measurement technique
allows 99.99% fidelity to be reached in 145us average detection time. For a
hyperfine qubit stored in the long-lived 4S_1/2 (F=3, F=4) sub-levels of 43Ca+
we propose and implement a simple and robust optical pumping scheme to transfer
the hyperfine qubit to the optical qubit, capable of a theoretical fidelity
99.95% in 10us. Experimentally we achieve 99.77(3)% net readout fidelity,
inferring at least 99.87(4)% fidelity for the transfer operation.
