The subject of this paper is the analysis and the design of a class of robust discrete-time variable-structure controllers of integral type. We refer to the discontinuous integral control (DIC) technique, already developed by the authors for the time-continuous case, where a nonlinear integral action is used for estimating the external disturbance and keeping small the discontinuous term in the control law and the subsequent chattering effect. A discretised version of the DIC algorithm, which is properly modified in order to take into account the time-derivative information on the controlled variable, is introduced and discussed. The so-obtained discrete-time controller shows to be very effective in reducing the chattering when the system parameters are known. Moreover, to cope with a specific case of unknown parameters, a new adaptive control structure is proposed and analysed. The local asymptotic stability of the resulting closed-loop system is proved in the case of constant external disturbances. Finally, simulation results on the velocity control of a DC motor end the paper.