Semantic communication, rather than on a bit-by-bit recovery of the transmitted messages, focuses on the meaning and the goal of the communication itself. In this paper, we propose a novel semantic image coding scheme that preserves the semantic content of an image, while ensuring a good trade-off between coding rate and image quality. The proposed Semantic-Preserving Image Coding based on Conditional Diffusion Models (SPIC) transmitter encodes a Semantic Segmentation Map (SSM) and a low-resolution version of the image to be transmitted. The receiver then reconstructs a high-resolution image using a Denoising Diffusion Probabilistic Models (DDPM) doubly conditioned to the SSM and the low-resolution image. As shown by the numerical examples, compared to state-of-the-art (SOTA) approaches, the proposed SPIC exhibits a better balance between the conventional rate-distortion trade-off and the preservation of semantically-relevant features. Code available at https://github.com/frapez1/SPIC