Sensor networks are gradually moving towards full-IPv6 architectures and play an important role in the upcoming Internet of Things. Some mission-critical applications of sensor networks will require a level of reliability that excludes the presence of single points of failure, as it is often the case today for the gateways connecting sensor networks to the Internet. In this paper, we introduce RPL-6LBR, a 6LoWPAN border router that addresses mission-critical deployments through redundancy. The paper discusses how existing standards may be leveraged to enable redundant border router synchronization, while identifying certain of their shortcomings. We also propose innovative network architectures incorporating multiple border routers, which deal with redundancy and node mobility without requiring any synchronization among the border routers. We implement the proposed RPL-6LBR in the Contiki operating system and report on this implementation through trials on a small-scale testbed and simulator. Our results open new possibilities for real-world wireless sensor networks requiring reliable border routers, and guide further standardization efforts in emerging technologies in support of the Internet of Things.