Co-authored by a Computer Scientist and a Digital Humanist, this article examines the challenges faced by cultural herit- age institutions in the digital age, which have led to the closure of the vast majority of born-digital archival collections. It focuses particularly on cultural organizations such as libraries, museums and archives, used by historians, literary scholars and other Humanities scholars. Most born-digital records held by cultural organizations are inaccessible due to privacy, copyright, commercial and technical issues. Even when born-digital data are publicly available (as in the case of web archives), users often need to physically travel to repositories such as the British Library or the Bibliothèque Nationale de France to consult web pages. Provided with enough sample data from which to learn and train their models, AI, and more specifically machine learning algorithms, offer the opportunity to improve and ease the access to digital archives by learn- ing to perform complex human tasks. These vary from providing intelligent support for searching the archives to automate tedious and time-consuming tasks. In this article, we focus on sensitivity review as a practical solution to unlock digital archives that would allow archival institutions to make non-sensitive information available. This promise to make archives more accessible does not come free of warnings for potential pitfalls and risks: inherent errors, "black box" approaches that make the algorithm inscrutable, and risks related to bias, fake, or partial information. Our central argument is that AI can deliver its promise to make digital archival collections more accessible, but it also creates new challenges - particularly in terms of ethics. In the conclusion, we insist on the importance of fairness, accountability and transparency in the process of making digital archives more accessible.