Successfully adopting AI and realising its full innovation potential requires different competences within a company. We identified five clusters, namely, AI decision-making, AI utilisation, AI foundational, AI development and leadership & moderation competences, as the basis for our AI competence framework, combining 35 individual competences. Based on a quantitative survey of 215 companies, we determined the importance of these competences for the successful adoption of AI innovations and their current availability within companies. According to our findings, AI foundational competences play a particularly critical role compared to the other competence clusters, which are considered important but comparatively rarely available. Furthermore, our analyses show that companies with higher levels of AI foundational, AI development, and AI utilisation competences have significantly higher AI innovation capabilities. Again, in particular AI foundational competences seem to fertilize the capabilities to identify appropriate AI use cases, to make decisions about AI innovation adoption, to successfully integrate AI into internal processes, and to use the AI innovation effectively within the organisation. Our findings thus enrich the theoretical discourse on competences for organisational adoption of AI innovations and guide practitioners in taking action to develop the necessary competences.