Recent experimental breakthroughs in satellite quantum communications have opened up the possibility of creating a global quantum internet using satellite links. This approach appears to be particularly viable in the near term, due to the lower attenuation of optical signals from satellite to ground, and due to the currently short coherence times of quantum memories. These drawbacks prevent ground-based entanglement distribution using atmospheric or optical-fiber links at high rates over long distances. In this work, we propose a global-scale quantum internet consisting of a constellation of orbiting satellites that provides a continuous on-demand entanglement distribution service to ground stations. The satellites can also function as untrusted nodes for the purpose of long-distance quantum-key distribution. We determine the optimal resource cost of such a network for obtaining continuous global coverage. We also analyze the performance of the network in terms of achievable entanglement-distribution rates and compare these rates to those that can be obtained using ground-based quantum-repeater networks.