This study is concerned with a singularly perturbed three-point boundary-value problem with delay. Firstly, bounds on the solution and its derivative of the solution to be used later in the paper are obtained. To solve it numerically, we use an exponentially fitted difference scheme on an equidistant mesh which is established by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. Then, the stability and convergence analysis of difference scheme is given and it is uniformly convergent in perturbation parameter. Furthermore, numerical results which show the effectiveness of the method are presented.