For a linear system driven by a prescribed-time stabilizing control algorithm from Song et al. (2017), in the presence of matched disturbances and measurement noises, an implicit Euler discretization scheme is presented and analyzed. The discretized version of the closed-loop system demonstrates a uniform fixed-time convergence as the continuous-time counterpart. Moreover, the discretized dynamics are robustly stable with respect to the measurement noise with a linear gain (the property that does not exist in the continuous time). It is proposed to apply the control recursively on infinite horizon rather than to be confined to a prescribed-time interval. Finally, a sampled-and-hold implementation of the prescribed-time stabilizing control is considered using the implicitly discretized closed-loop system as the tracking objective. The efficiency of the suggested sampled realization of the control is illustrated through numeric experiments.