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In recent years, Denoising Diffusion Probabilistic Models (DDPMs) have demonstrated exceptional performance in various 2D generative tasks. Following this success, DDPMs have been extended to 3D shape generation, surpassing previous methodologies in this domain. While many of these models are unconditional, some have explored the potential of using guidance from different modalities. In particular, image guidance for 3D generation has been explored through the utilization of CLIP embeddings. However, these embeddings are designed to align images and text, and do not necessarily capture the specific details needed for shape generation. To address this limitation and enhance image-guided 3D DDPMs with augmented 3D understanding, we introduce CISP (Contrastive Image-Shape Pre-training), obtaining a well-structured image-shape joint embedding space. Building upon CISP, we then introduce IC3D, a DDPM that harnesses CISP's guidance for 3D shape generation from single-view images. This generative diffusion model outperforms existing benchmarks in both quality and diversity of generated 3D shapes. Moreover, despite IC3D's generative nature, its generated shapes are preferred by human evaluators over a competitive single-view 3D reconstruction model. These properties contribute to a coherent embedding space, enabling latent interpolation and conditioned generation also from out-of-distribution images. We find IC3D able to generate coherent and diverse completions also when presented with occluded views, rendering it applicable in controlled real-world scenarios.