The most demanding tenants of shared clouds require complete isolation from their neighbors, in order to guarantee that their application performance is not affected by other tenants. Unfortunately, while shared clouds can offer an option whereby tenants obtain dedicated servers, they do not offer any network provisioning service, which would shield these tenants from network interference. In this paper, we introduce Links as a Service, a new abstraction for cloud service that provides physical isolation of network links. Each tenant gets an exclusive set of links forming a virtual fat tree, and is guaranteed to receive the exact same bandwidth and delay as if it were alone in the shared cloud. Under simple assumptions, we derive theoretical conditions for enabling LaaS without capacity over-provisioning in fat-trees. New tenants are only admitted in the network when they can be allocated hosts and links that maintain these conditions. Using experiments on real clusters as well as simulations with real-life tenant sizes, we show that LaaS completely avoids the performance degradation caused by traffic from concurrent tenants on shared links. Compared to mere host isolation, LaaS can improve the application performance by up to 200%, at the cost of a 10% reduction in the cloud utilization.