In this paper, we investigate theoretically the excitation of modes by a line source, as well as their modal structure and dispersion, in a parallel-plate waveguide that is filled with a pair of parallel slabs; one being a lossless "double-negative (DNG)" material and the other being a lossless conventional "double-positive (DPS)" medium. Previously, we have shown that such "conjugate" pairing of DNG and DPS materials may lead to reduction of size in one-dimensional cavity resonators and waveguides. The analysis presented here describes the proper modes in such a 1-D parallel plate waveguide, and reveals the possibility of no cut-off thickness for this class of parallel-plate waveguides, which implies that a proper TE or TM mode may always be excited in such a waveguide independent of the overall thickness of the waveguide. The excitation of guided modes by a line source and their power-flow peculiarities are discussed, and the analogy to the concept of "open 1-D cavity" in such a waveguide is mentioned. Some of the interesting features and physical insights regarding the power flow in these guided modes will be presented.