We study TE-wave propagation in a circular waveguide with a graded transition from a lossy right-handed material (RHM) filling the left-hand half of the waveguide to the impedance-matched lossy left-handed material (LHM) filling the right-hand half of the waveguide. We obtain exact analytical solutions to Maxwell's equations and perform corresponding numerical simulations in COMSOL. An excellent agreement is obtained between the numerical simulations and analytical results. The presented method can model smooth realistic material transitions, where the interface width is an additional degree of freedom in the design of practical RHM-LHM interfaces.