This paper concerns a global stabilization control problem for a two-link underactuated robot moving in the vertical plane with a single actuator at the first joint and a spring between the two links (flexible elbow joint). Different from existing literature, we obtain the following results. First, we present a property about the angle of the first joint, the torque and the equilibrium configuration of the robot which is critical for the motion analysis of the robot in this paper. Second, we prove that when the spring constant is bigger than a mechanical parameter related to a coefficient of a gravitational term, the PD control on the angle of the first joint can globally stabilize the robot at the upright equilibrium point, where two links are in the upright position. Finally, we validate the presented theoretical results via numerical investigation.