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Dataset distillation (DD) compresses large datasets into smaller ones while preserving the performance of models trained on them. Although DD is often assumed to enhance data privacy by aggregating over individual examples, recent studies reveal that standard DD can still leak sensitive information from the original dataset due to the lack of formal privacy guarantees. Existing differentially private (DP)-DD methods attempt to mitigate this risk by injecting noise into the distillation process. However, they often fail to fully leverage the original dataset, resulting in degraded realism and utility of the distilled dataset. This paper introduces DP-GenG, a novel framework that addresses the key limitations of current DP-DD by leveraging DP-generated data. Specifically, DP-GenG initializes the distilled dataset with DP-generated data to enhance realism. Then, guided by this data, it refines the DP-feature matching technique to distill the original dataset under a small privacy budget, and trains an expert model to align the distilled examples with their class distribution. Furthermore, we design a privacy budget allocation strategy to determine budget consumptions across DP components and provide a theoretical analysis of the overall privacy guarantees. Extensive experiments show that DP-GenG significantly outperforms state-of-the-art DP-DD methods in terms of both dataset utility and robustness against membership inference attacks, establishing a new paradigm for privacy-preserving dataset distillation.