The authors consider a scalar linear system of Itô stochastic delay differential equations \[ \begin{cases} dy(t) & = (a y(t) + b y(t -\tau) dt + (c y (t) + dy(t-\tau))dW(t), \quad t \geq 0,\\ y(t) & = \psi (t), \quad t\in [-\tau, 0]\end{cases}\tag{1} \] where \(W(t)\) is on dimensional standard Wiener process, \(\tau >0.\) A split-step backward Euler (SSBE) scheme for solving this system is constructed. The authors constructed the SSBE method by \( Y_k = \psi (kh),\) when \(k=-m, -m+1, \dots, 0\), \(h=t \over N\) and when \(k \geq 0\) \[ \begin{cases} Y_{k}^* & = Y_k + h[a Y_k^* +b Y_{k-m+1}],\\ Y_{k+1} & = Y_k^* + (c Y_k^* +d Y_{k-m+1}) \Delta W_k \end{cases} \] where \(Y_k\) is the numerical approximation of \(y(t_k)\) with \(t_k =kh.\) The following theorem is the main result of this paper. Theorem: Assume the condition \(a0\) then the SSBE methods is MS-stable and the stepsize satisfies \(h \in (0, h_1 (a, b, c, d)),\) where \[ h_1(a,b ,c,d)= \frac{-[2a +2|b| + (|c| +|d|)^2]}{4|b| c^2 + b^2 - a^2}. \] \item[(iii)] if \(ad -bc \neq 0\) then the SSBE methods is MS-stable and the stepsize satisfies \(h \in (0, h_2 (a, b, c, d)),\) where \[ h_2(a,b ,c,d)= \frac{-[2|b|c^2 -2a |cd| + b^2 - 2ad^2 + 2bcd -a^2] +\sqrt\Delta}{2(ad - bc)^2}. \] \end{itemize}} Here, \[ \Delta = [2|b|c^2 - 2a|cd| +b^2 -2ad^2 +2bcd -a^2]^2 - 4(ad-bc)^2 [2a + 2|b| +( |c| +|d|)^2]. \] Several illustrative numerical examples of applying the SSBE method are presented.