A technique protects against the failure of a border router between two domains in a computer network using Fast Reroute and backup tunnels. According to the technique, the protected border router advertises a list of all its adjacent next-hop routers (i.e., its “neighbors”). A neighbor in the first domain that is immediately upstream to the protected border router and that is configured to protect the border router (i.e., the “protecting router”) selects a neighbor in a second domain (i.e., a “next-next-hop,” NNHOP) to act as a “merge point” of all the NNHOPs of that domain. The protecting router calculates a backup tunnel to the merge point that excludes the protected border router and associates the backup tunnel with all “protected prefixes.” The merge point then “stitches” additional backup tunnels onto the backup tunnel to provide a stitched tunnel to each remaining NNHOP. When the protected border router fails, Fast Reroute is triggered, and all protected prefix traffic is rerouted onto the backup tunnel to the merge point, which either forwards the traffic to its reachable prefixes or to a corresponding stitched tunnel.