Many studies show that, when Internet links go up or down, the dynamics of BGP may cause several minutes of packet loss. The loss occurs even when multiple paths between the sender and receiver domains exist, and is unwarranted given the high connectivity of the Internet. Our objective is to ensure that Internet domains stay connected as long as the underlying network is connected. Our solution, R-BGP works by pre-computing a few strategically chosen failover paths. R-BGP provably guarantees that a domain will not become disconnected from any destination as long as it will have a policy-compliant path to that destination after convergence. Surprisingly, this can be done using a few simple and practical modifications to BGP, and, like BGP, requires announcing only one path per neighbor. Simulations on the AS-level graph of the current Internet show that R-BGP reduces the number of domains that see transient disconnectivity resulting from a link failure from 22% for edge links and 14% ...