  We reconsider the Eigen's quasi-species model for competing self-reproductive
macromolecules in populations characterized by a single-peaked fitness
landscape. The use of ideas and tools borrowed from polymers theory and
statistical mechanics, allows us to exactly solve the model for generic DNA
lengths d. The mathematical shape of the quasi-species confined around the
master sequence is perturbatively found in powers of 1/d at large d. We
rigorously prove the existence of the error-threshold phenomenon and study the
quasi-species formation in the general context of critical phase transitions in
physics. No sharp transitions exist at any finite d, and at $d\to \infty$ the
transition is of first order. The typical r.m.s. amplitude of a quasi-species
around the master sequence is found to diverge algebraically with exponent
$\nu_{\perp}=1$ at the transition to the delocalized phase in the limit $d\to
\infty$.
