Recently, diffusion models have emerged as a promising paradigm for molecular design and optimization. However, most diffusion-based molecular generative models focus on modeling 2D graphs or 3D geometries, with limited research on molecular sequence diffusion models. The International Union of Pure and Applied Chemistry (IUPAC) names are more akin to chemical natural language than the Simplified Molecular Input Line Entry System (SMILES) for organic compounds. In this work, we apply an IUPAC-guided conditional diffusion model to facilitate molecular editing from chemical natural language to chemical language (SMILES) and explore whether the pre-trained generative performance of diffusion models can be transferred to chemical natural language. We propose DiffIUPAC, a controllable molecular editing diffusion model that converts IUPAC names to SMILES strings. Evaluation results demonstrate that our model outperforms existing methods and successfully captures the semantic rules of both chemical languages. Chemical space and scaffold analysis show that the model can generate similar compounds with diverse scaffolds within the specified constraints. Additionally, to illustrate the model's applicability in drug design, we conducted case studies in functional group editing, analogue design and linker design.