This paper deals with the stabilization problem of the rotating disk-beam system with infinite memory in the moment control. A linear feedback law is proposed, which consists of a torque control and a moment control with an infinite memory term. It is shown that the closed-loop system is well-posed and exponentially stable provided that the desired constant angular velocity and the memory kernel function satisfy certain conditions. The main ingredient of the proof is the frequency-domain method. Numerical simulations are also put forward to validate the results.