A continuous beam scanning leaky-wave antenna (LWA) supporting lateral radiation is proposed in this paper. It is based on a spoof surface plasmon (SSP) transmission line (TL) which is a single-layer conductor structure. The proposed SSP TL is slotted periodically with rectangular holes on the metallic strip conductor. Circular patches are loaded periodically along one side of the SSP TL to achieve an additional momentum for converting the slow-wave mode to a fast-wave mode and then modulate the SSP wave into a spatial wave. The proposed SSP LWA produces a single lateral beam rather than double beams usually existed in conventional SSP LWAs, as highly desired in many applications. As operating frequencies increase from 7 to 15 GHz, the proposed SSP LWA achieves a continuous beam scanning range of 65° from forward to backward with stable high gains. As its primary advantage, this proposed SSP LWA does not lose radiation efficiency even at broadside, which is often suffered by conventional LWAs. Both the transmission characteristics of the SSP TL and the radiation mechanism of the SSP LWA are investigated. Finally, the proposed SSP LWA is designed, fabricated, and measured. The measured results agree well with the simulated ones over a wide operating frequency range, which validates its good performances. The proposed SSP LWAs provide a low-cost solution to planarly integrated communication systems.