Dynamical chiral-symmetry breaking in asymptotically free theories with a slowly running coupling is analyzed. When the confinement scale Λ is much less than the cutoff M beyond which the theory cannot be used in isolation, the dynamical mass Σ(p) starts from a value ~=Λ for momenta p<~Λ and falls slowly for a significant range p>Λ. It then takes on the asymptotic form (lnp)a/p2 where a>1. In hypercolor theories this behavior generates sufficiently large fermion masses for a higher value of M than naively expected. This in turn could allow for an adequate suppression of flavor-changing neutral currents.