In this paper, a singularly perturbed convection–diffusion equation is studied. At first, the original problem is transformed into a parameterized singularly perturbed Volterra integro-differential equation by using an integral transform. Then, a second-order finite difference method on an arbitrary mesh is given. The stability and local truncation error estimates of the discrete schemes are analyzed. Based on the mesh equidistribution principle and local truncation error estimation, an adaptive grid algorithm is given. In addition, in order to calculate the parameters of the transformation equation, a nonlinear unconstrained optimization problem is constructed. Numerical experiments are given to illustrate the effectiveness of our presented adaptive grid algorithm.