  The quantum error threshold is the highest (model-dependent) noise rate which
we can tolerate and still quantum-compute to arbitrary accuracy. Although noise
thresholds are frequently estimated for the Steane seven-qubit, distance-three
quantum code, there has been no proof that a constant threshold even exists for
distance-three codes. We prove the existence of a constant threshold. The
proven threshold is well below estimates, based on simulations and analytic
models, of the true threshold, but at least it is now known to be positive.
