  Quantum information processors promise fast algorithms for problems
inaccessible to classical computers. But since qubits are noisy and
error-prone, they will depend on fault-tolerant quantum error correction
(FTQEC) to compute reliably. Quantum error correction can protect against
general noise if -- and only if -- the error in each physical qubit operation
is smaller than a certain threshold. The threshold for general errors is
quantified by their diamond norm. Until now, qubits have been assessed
primarily by randomized benchmarking, which reports a different "error rate"
that is not sensitive to all errors, and cannot be compared directly to diamond
norm thresholds. Here we use gate set tomography (GST) to completely
characterize operations on a trapped-Yb$^+$-ion qubit and demonstrate with very
high ($>95\%$) confidence that they satisfy a rigorous threshold for FTQEC
(diamond norm $\leq6.7\times10^{-4}$).
