An H¹ space-time discontinuous Galerkin (STDG) scheme for convectiondiffusion equations in one spatial dimension is constructed and analyzed. This method is formulated by combining the H¹ Galerkin method and the space-time discontinuous finite element method that is discontinuous in time and continuous in space. The existence and the uniqueness of the approximate solution are proved. The convergence of the scheme is analyzed by using the techniques in the finite difference and finite element methods. An optimal a-priori error estimate in the L∞(H¹ ) norm is derived. The numerical experiments are presented to verify the theoretical results.