We study the problem of differentially private clustering under\ninput-stability assumptions. Despite the ever-growing volume of works on\ndifferential privacy in general and differentially private clustering in\nparticular, only three works (Nissim et al. 2007, Wang et al. 2015, Huang et\nal. 2018) looked at the problem of privately clustering "nice" k-means\ninstances, all three relying on the sample-and-aggregate framework and all\nthree measuring utility in terms of Wasserstein distance between the true\ncluster centers and the centers returned by the private algorithm. In this work\nwe improve upon this line of works on multiple axes. We present a far simpler\nalgorithm for clustering stable inputs (not relying on the sample-and-aggregate\nframework), and analyze its utility in both the Wasserstein distance and the\nk-means cost. Moreover, our algorithm has straight-forward analogues for "nice"\nk-median instances and for the local-model of differential privacy.\n