A new scheme that solves the advection-diffusion equation for tracers in a spectral General Circulation Model (GCM) is presented. The main ideas are (1) using a monotonic and smooth functional of the tracer as prognostic variable to ensure positive definite concentrations and continuity of all derivatives and (2) defining an adjustable tracer-mass correction as a multiplication of the tracer in grid space, giving rise to an efficient correction in spectral space. Common standard benchmark tests for two-dimensional horizontal advection using deformational wind fields show that the new scheme is accurate and essentially not diffusive. A three-dimensional test is proposed in order to validate vertical transport. Additionally to standard error norms and global tracer mass, the entropy of mixing is introduced as another conservation constraint and utilized to determine the strength of the mass correction which is a free parameter. The transport scheme is applied in a mechanistic spectral GCM from the surface to the lower thermosphere. It is extended such that the mass correction takes the diffusion and other nonconservative effects explicitly into account. By this method we estimate the mean age of air along with its dependence on the turbulent horizontal Schmidt number.