  Perfect cloning of a known set of states with arbitrary prior probabilities
is possible if we allow the cloner to sometimes fail completely. In the optimal
case the probability of failure is at its minimum allowed by the laws of
quantum mechanics. Here we show that it is possible to lower the failure rate
below that of the perfect probabilistic cloner but the price to pay is that the
clones are not perfect; the global fidelity is less than one. We determine the
optimal fidelity of a cloner with a Fixed Failure Rate (FFR cloner) in the case
of a pair of known states. Optimality is shown to be attainable by a
measure-and-prepare protocol in the limit of infinitely many clones. The
optimal protocol consists of discrimination with a fixed rate of inconclusive
outcome followed by preparation of the appropriate clones. The convergence
shows a symmetry-breaking second-order phase transition in the fidelity of the
approximate infinite clones.
