This paper presents an energy-bounding approach for robustly stable bilateral teleoperation over a communication channel with severe variable time delays and packet drops. We extend the energy-bounding algorithm (EBA) for haptic interaction with virtual environments to bilateral teleoperation with remote environments by using an analogy between haptic interaction and teleoperation controls. Robust stability is achieved by both restricting the extra energy that is generated by the sample-and-hold to within the consumable energy in the master device or slave robot and passifying the communication network. Theoretical analyses of transparency are performed for both position and force tracking aspects. Comprehensive test results for various free and contact motions subsequently show that the proposed bilateral EBA can ensure robust stability against fairy large constant/variable round trip time delays (tested for up to 5 sec for free motion and 600 msec for contact motion within the device workspace) as well as for packet losses of up to 90% during data transmission.