Mesh generation is of great value in various applications involving computer graphics and virtual content, yet designing generative models for meshes is challenging due to their irregular data structure and inconsistent topology of meshes in the same category. In this work, we design a novel sparse latent point diffusion model for mesh generation. Our key insight is to regard point clouds as an intermediate representation of meshes, and model the distribution of point clouds instead. While meshes can be generated from point clouds via techniques like Shape as Points (SAP), the challenges of directly generating meshes can be effectively avoided. To boost the efficiency and controllability of our mesh generation method, we propose to further encode point clouds to a set of sparse latent points with point-wise semantic meaningful features, where two DDPMs are trained in the space of sparse latent points to respectively model the distribution of the latent point positions and features at these latent points. We find that sampling in this latent space is faster than directly sampling dense point clouds. Moreover, the sparse latent points also enable us to explicitly control both the overall structures and local details of the generated meshes. Extensive experiments are conducted on the ShapeNet dataset, where our proposed sparse latent point diffusion model achieves superior performance in terms of generation quality and controllability when compared to existing methods.