To provide wireless Internet access, WiFi networks have been deployed in many regions such as buildings and campuses. However, WiFi networks are still insufficient to support ubiquitous wireless service due to their narrow coverage. One possibility to resolve this deficiency is to integrate WiFi networks with the wide-range WiMAX networks. Under such an integrated WiMAX and WiFi network, how to conduct energy-efficient handovers is a critical issue. In this paper, we propose a handover scheme with geographic mobility awareness (HGMA), which considers the historical handover patterns of mobile devices. HGMA can conserve the energy of handovering devices from three aspects. First, it prevents mobile devices from triggering unnecessary handovers according to their received signal strength and moving speeds. Second, it contains a handover candidate selection method for mobile devices to intelligently select a subset of WiFi access points or WiMAX relay stations to be scanned. Therefore, mobile devices can reduce their network scanning and thus save their energy. Third, HGMA prefers mobile devices staying in their original WiMAX or WiFi networks. This can prevent mobile devices from consuming too much energy on interface switching. In addition, HGMA prefers the low-tier WiFi network over the WiMAX network and guarantees the bandwidth requirements of handovering devices. Simulation results show that HGMA can save about 59- 80% of energy consumption of a handover operation, make mobile devices to associate with WiFi networks with 16-62% more probabilities, and increase about 20-61% of QoS satisfaction ratio to handovering devices.