A method and apparatus for increasing the success rate of a handover of a connected mode UE while minimizing the power consumption of the UE during DRX operation in a mobile communication system is provided. The method includes the steps of transmitting a cell change report to the serving base station, and then suspending the DRX operation to transition to an active mode; receiving a handover command message, and completing the handover to a target cell; checking if the handover command message includes a DRX resume command; waiting until a control message including the DRX resume command is received from the target base station; and after the control message including the DRX resume command is received, transitioning from the active mode to a sleep mode, and resuming the DRX operation.