We have developed an analytical self-consistent-field (SCF) theory describing conformations of weakly charged polyelectrolyte chains tethered to the solid-liquid interface and immersed in a solution of low molecular weight salt. Depending on the density of grafting of the polyelectrolytes to the interface and on the salt concentration we distinguish three main asymptotic regimes of behavior of the grafted layer. These regimes are characterized by (i) unscreened Coulomb repulsion between polyions, (ii) screening of the interchain interactions predominantly by counterions, or (iii) screening ensured by co-ions and counterions of the salt, respectively. We have demonstrated that all the structural and thermodynamic properties of the brush are determined by two dimensionless parameters, i.e., the bare Gouy-Chapman length normalized by the characteristic brush thickness and the bulk Debye screening length, respectively. The theory describes contraction of the brush as a whole and its internal structural rearrangements with increasing salt concentration. In particular, we consider variation of the polymer density profile and distributions of the end segments and small ions with increasing salt concentration. The maximum sensitivity of the brush to the addition of salt is predicted in the intermediate range of grafting density while dense and sparse brushes are less affected by added salt.