The Acoustic Black Hole (ABH) embedded in a beam is a termination that provides effective vibration absorption. The ABH effect, usually achieved by adding a visco-elastic coating in the flexible part of the termination, results in a reflection coefficient that varies with frequency, with local maxima and minima associated with local modes trapped in the termination. The density of these modes induces narrow absorption bands whose overlap makes the ABH very efficient from a cut-on frequency. In this paper, a new way to take advantage of the ABH effect is developed analytically, numerically and experimentally by applying a localised damping to the termination by means of elastic and damping point supports. This new type of device provides a new way to control the ABH effect inducing low reflection coefficient values over wide and low frequency ranges.