In this paper, we propose a single-port dual-beam leaky-wave antenna (LWA) in the terahertz (THz) band based on a composite spoof surface plasmon polariton (SSPP) waveguide. The antenna can generate three independent transmission channels by exciting two independent modes inherent to hole and groove structures, respectively. By periodic modulation of the hole and groove structures, we achieve dual-beam scanning through a broad radiation angle using only the -1st space harmonics of the two modes, hence avoiding the instability of the -2rd space harmonic. Within the operating frequency range of 0.62—0.85 THz, the gain ranges from 13.5 dBi to 17 dBi for the backward beam, and from 6 dBi to 11.8 dBi for the forward beam. The antenna can accomplish continuous backward beam through broadside to forward beam scanning with a total scanning range of 116° and an average efficiency of about 92%. The antenna exhibits a great potential in the design of multi-transceiver radar system in the THz band and multi-beam LWAs.