Cubic B-spline Galerkin method, based on second and fourth order single step methods for time integration is used to solve numerically the advection diffusion equation (ADE). Second order single step method is also known as Crank Nicolson method. Two numerical examples are used to validate the proposed method which is found to be accurate and efficient. The effects of the advection and diffusion terms on the solution domain and the absolute error of the numerical solution are studied with the help of graphs. The obtained results show that the proposed fourth order single step method has a high success as a numerical technique for solving the ADE.