We experimentally and theoretically investigate the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide (PPWG) for the propagation of broadband THz pulses. We demonstrate undistorted THz pulse propagation via the single TE1 mode, solving the group-velocity-dispersion and spectral-filtering problems caused by the mode's low-frequency cutoff. We observe a remarkable counterintuitive property of the TE1 mode: its attenuation decreases with increasing frequency for all frequencies above cutoff. This phenomenon has not been observed with any other THz waveguide to date, and it can enable extremely low-loss propagation. We present a physical interpretation of this frequency-dependent behavior using a simple plane-wave description of the TE1 mode propagation. We also find that it is possible to achieve almost 100% coupling to the TE1 mode from a focused free-space Gaussian beam. In addition, using the above plane-wave analysis, we show how to mitigate the diffraction losses inherent to long path-length PPWGs via the use of transverse-concave plates.