  A new diagrammatic method, which is a reformulation of Berezinskii's
technique, is constructed to study the density of electronic states
$\rho(\epsilon,\phi)$ of a one-channel weakly disordered ring, threaded by an
external magnetic flux. The exact result obtained for the density of states
shows an oscillation of $\rho(\epsilon,\phi)$ with a period of the flux quantum
$\phi_0= hc/e $. As the sample length (or the impurity concentration) is
reduced, a transition takes place from the weak localization regime (L >> l) to
the ballistic one (L <= l). The analytical expression for the density of states
shows the exact dependence of $\rho(\epsilon,\phi)$ on the ring's circumference
and on disorder strength for both regimes.
