We present a high-order Galerkin method in both space and time for the one-dimensional unsteady advectiondiffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate and efficient.The numerical results indicate that the symmetric version of IPDG is typically more accurate and more ecient compared to the other approaches.