An implicit Euler discretization scheme is given for a linear system driven by the prescribed-time stabilizing control algorithm from [1] in the presence of matched disturbances and measurement noise. The discretized version preserves all main properties of the continuous-time counterpart, and can be recursively applied on infinite horizon rather than confined to the prescribed-time interval. In addition, the discretized estimation error is robustly stable with respect to the measurement noise with a linear gain. The efficiency of the suggested discretization is illustrated through numeric experiments.