An implicit Euler discretization scheme of the prescribed-time converging observer from Holloway and Krstic, (2019) is proposed, which preserves all main properties of the continuous-time counterpart, and it is investigated how to apply recursively this discretization on any interval of time. In addition, it is demonstrated that the estimation error is robustly stable with respect to the measurement noise with a linear gain (the property that does not hold in the continuous-time setting). The efficiency of the suggested discrete-time observer is illustrated through numeric experiments.