The traditional passive absorber is fully effective within a narrow and certain frequency band. To solve this problem, a time-delayed acceleration feedback is introduced to convert a passive absorber into an active one. Both the inherent and the intentional time delays are included. The former mainly comes from signal acquiring and processing, computing, and applying the actuation force, and its value is fixed. The latter is introduced in the controller, and its value is actively adjustable. Firstly, the mechanical model is established and the frequency response equations are obtained. The regions of stability are delineated in the plane of control parameters. Secondly, the design scheme of control parameters is performed to help select the values of the feedback gain and time delay. Thirdly, the experimental studies are conducted. Effects of both negative and positive feedback control are investigated. Experimental results show that the proper choices of control parameters may broaden the effective frequency band of vibration absorption. Moreover, the time-delayed absorber greatly suppresses the resonant response of the primary system when the passive absorber totally fails. The experimental results are in good agreement with the theoretical predictions and numerical simulations.