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This study presents the first differentially private and fair clustering method, built on the recently proposed density-based fair clustering approach. The method addresses the limitations of fair clustering algorithms that necessitate the use of sensitive personal information during training or inference phases. Two novel solutions, the Gaussian mixture density function and Voronoi cell, are proposed to enhance the method's performance in terms of privacy, fairness, and utility compared to previous methods. The experimental results on both synthetic and real-world data confirm the compatibility of the proposed method with differential privacy, achieving a better fairness-utility trade-off than existing methods when privacy is not considered. Moreover, the proposed method requires significantly less computation time, being at least 3.7 times faster than the state-of-the-art.