Large Language Models (LLMs) memorize, and thus, among huge amounts of uncontrolled data, may memorize Personally Identifiable Information (PII), which should not be stored and, consequently, not leaked. In this paper, we introduce Private Memorization Editing (PME), an approach for preventing private data leakage that turns an apparent limitation, that is, the LLMs' memorization ability, into a powerful privacy defense strategy. While attacks against LLMs have been performed exploiting previous knowledge regarding their training data, our approach aims to exploit the same kind of knowledge in order to make a model more robust. We detect a memorized PII and then mitigate the memorization of PII by editing a model knowledge of its training data. We verify that our procedure does not affect the underlying language model while making it more robust against privacy Training Data Extraction attacks. We demonstrate that PME can effectively reduce the number of leaked PII in a number of configurations, in some cases even reducing the accuracy of the privacy attacks to zero.