Service-based architecture in the 5G core network, disaggregated RAN, and network slicing are a few of the 5G network's innovative paradigms. There are however new requirements for 6G, such as the efficient use of integrated cloud resources, end-to-end network programmability, and the dynamic integration of multiple connectivity domains to realize intelligent connectivity across increasingly pervasive cloud environments. The 6G-Cloud project will research, develop, and validate key technologies to realize an artificial intelligence (AI)-native and cloud-friendly system architecture atop the cloud continuum. It will integrate cloud resources offered by multiple stakeholders and allow network functions from different 6G network segments to be composed flexibly and dynamically based on service needs in hybrid cloud environments. 6G-Cloud will address the following key topics: i) designing a holistic, AI-native service-oriented 6G system architecture atop a cloud continuum; ii) developing AI-driven cloud continuum and management frameworks and business interfaces for a multistakeholder environment; iii) defining an AI/machine learning (ML) framework for native-AI support in the 6G system; iv) supporting the 6G "network-of-networks" concept using service-oriented network design. 6G-Cloud will incorporate scalability, sustainability, resilience, and security requirements into system design. The concept will be validated by three well-defined proofs-of-concept and promoted through 6G architectural standardization work and open-source projects to reach maximum impact.