The launch of the generative artificial intelligence (gen AI) application ChatGPT by OpenAI launched artificial intelligence into public discourse and led to a wave of mass uptake of this technology in organizations in the private sector. At the same time, AI is increasingly incorporated into government functions and the public sector. We propose that governments and the public sector can set an example for the responsible use of AI technologies by following the principles of algorithmic governance traditionally recommended to the private sector. Algorithmic governance has historically been defined in the literature as governance by algorithms, or how artificial intelligence is used to make governance decisions and affect social ordering. However, we take an alternative approach; instead, we conceptualize algorithmic governance as the governance of algorithms. We summarize the risks of generative AI use in governments and the public sector, then outline algorithmic governance principles, a step-by-step approach to implementing algorithmic governance into government or public sector projects, opportunities for inter-sector collaboration, and conclusions.