Conformal antennas have been widely used in many fields due to their advantages of low air resistance and better visual appearance. In this paper, an arced conformal leaky-wave antenna (LWA) for a designable directional beam is proposed. The antenna is achieved based on a substrate-integrated waveguide (SIW). On the upper surface, a series of non-uniform transverse slots are etched. In order to guide the design of the antenna, as another key contribution of this work, a theoretical model for the traveling-wave structure is established. Using the model, the radiation property of the LWA is analyzed. In addition, by inputting the desired beam direction, the structural parameters of the LWA can be generated through the model. To verify the performance of the antenna and the model, an LWA prototype working at 28 GHz was fabricated and tested in a microwave anechoic chamber. The experimental results are in good agreement with the simulation results. The antenna achieved a gain of 9.96 dBi with cambered surface area of 1.89 λ02. The proposed method may be a promising candidate for conformal wireless communication applications.