In this paper, a band-notched absorber with high notch-band-edge selectivity is proposed and investigated. It is formed by introducing a narrower and full reflectance band into a wider absorption band. For each absorbing unit cell, two independently magnetic and electric resonances with adjacent resonant frequencies are employed to develop the narrower and full reflectance band, while a dipole-shaped metal strip with a resister printed on one side of the supporting substrate and a ground plane is used to realize the wider absorption band. By retrieving relative equivalent circuits of each cell, the performance of the proposed absorber can be qualitatively analyzed. Simply through tuning the frequencies of the independently magnetic and electric resonances, the performance of the notched band can be adjusted with high notch-band-edge selectivity. A sample of an absorber with the notched band frequency from 8.2 to 9.8 GHz, good absorption band from 4.8 to 8.0 GHz and 10.2 to 16 GHz has been designed, fabricated, and measured. Good agreement among circuit analysis, simulation results, and measurement results is finally obtained.