The paper is concerned with the delay differential equation \[ y'(t)=My(t-\tau), \quad t\geq 0,\quad y(t)=\phi(t), \quad -\tau\leq t\leq 0, \tag{1} \] where \(\tau>0, M\in {\mathbb{C}}^{m\times m},\) and \(\phi\in C([-\tau,0], {\mathbb{C}}^m).\) Equation (1) is first reformulated as the abstract Cauchy problem and then the discretization of the latter is performed. The main result of the paper states that the discretization of the abstract Cauchy problem is an asymptotically stable process on the class \({\mathcal C}_3\) of delay differential equations which are asymptotically stable for a fixed delay \(\tau.\)