We study analytically the steady-state regime of a network of n error-prone self-replicating templates forming an asymmetric hypercycle and its error tail. We show that the existence of a master template with a higher noncatalyzed self-replicative productivity a than the error tail ensures the stability of chains in which m < n-1 templates coexist with the master species. The stability of these chains against the error tail is guaranteed for catalytic coupling strengths K on the order of a. We find that the hypercycle becomes more stable than the chains only if K is on the order of a2. Furthermore, we show that the minimal replication accuracy per template needed to maintain the hypercycle, the so-called error threshold, vanishes as square root of n/K for large K and N < or = 4.