This article explores the struggle for ‘digital sovereignty’ in the European Union. A seeming contradiction – the internet, after all, spans the globe – digital sovereignty is portrayed as the winning geo-economic formula to keep the EU secure, competitive, and democratic in the digital future. Approaching digital sovereignty as a discursive claim and analysing it through a case study of the European cloud project Gaia-X, we show that there is no singular understanding of digital sovereignty in the EU. Instead, we identify six markedly different conceptions across the domains of security, economy and rights. The article outlines three scenarios for how the digital sovereignty agenda may develop and thus shape the EU’s digital policy and the EU’s relations with the rest of the world: constitutional tolerance (where the conceptions co-exist), hegemony (where one conception dominates), or collapse (where the agenda falls apart due to inbuilt conceptual contradictions).