On the basis of the ab-initio Full-Potential Linearized Augmented Plane Wave method we calculated the electronic structure for a nine layer Ag slab covered by a Cl layer on the top and bottom of the slab. The geometrical arrangement corresponds to a simple overlayer model of the Ag(001) surface and a c(2 x 2)-Cl adsorbate. For comparison, also the clean Ag(001) slab and the free Cl monolayers were calculated with the same geometry. The distance of the Cl layer from the Ag surface layer was obtained by total energy minimization. The derived Ag-Cl bond length of 2.59 Angstrom agrees well with recent experimental results. For the work function a value of 5.9 eV was calculated. The main goal of our paper is the study of surface states, which for the clean Ag(001) surface occur for ($) over bar k-point ($) over bar X at approximately 3eV above E(F) (state A of s-like character), and close to E(F) (state B of p-like character). Upon adsorption of Cl, state B splits into two states, whereby one of them does not interact with the Cl adsorbate and therefore remains approximately at the same energetic position as for the clean Ag(001) surface. The second one interacts in a strongly bonding way and is found 4.25 eV below E(F) still retaining the p-character at the Ag surface atoms. The fate of state A is not so clear. Although for the Cl covered Ag surface we still find a state near 3 eV, its charge character is quite different from the original state A. There is a second state at 4.3 eV above E(F), which still shows the s-like character of state A, but is less localized in the Ag surface layer due to the antibonding interaction with Cl.