A second-order finite difference procedure is presented For the inverse determination of the thermal conductivity in a one-dimensional heat conduction domain. In this approach, the thermal conductivity of the material is reconstructed by using the available temperature data at discrete grid points. The accuracy of the computational algorithm is investigated. To confirm the validity of the numerical procedure, various comparative examples are presented. The close agreement between the current results and the exact solutions confirms that the proposed finite difference procedure is an effective technique for the inverse determination of thermal conductivity. The method is applicable for linear and nonlinear spatially dependent, as well as temperature dependent, thermal conductivities. In addition, a key feature of the present technique is that a priori knowledge of the functional form for the thermal conductivity is not required.