In this paper, the boundary-value problem for a parameter dependent linear first order delay differential equation is analyzed. A finite difference method for approximate solution of this problem is presented. The method is based on fitted difference scheme on a uniform mesh which is achieved by using the method of integral identities which includes the exponential basis functions and applying interpolating quadrature formulas which contain the remainder term in integral form. Also, the method is proved first-order convergent in the discrete maximum norm. Moreover, a numerical example is solved using both the presented method and the Euler method and compared the computed results.