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We present a new design of a dual-band frequency selective surface (FSS) with closely spaced bands of operation and a highly-selective frequency response at each band. A multi-stage design procedure is also proposed for the design and synthesis of this class of frequency selective surfaces. The design procedure is based on synthesizing the desired device from its equivalent circuit parameter values. An approximate analytical technique is provided, which can be used to determine the values of the equivalent circuit parameters of this dual-band device from the basic system level characteristics of its transfer function including the center frequencies of operations of its two bands of operation and the bandwidth at each of these bands. The use of this design procedure is described in detail by a design example which follows the proposed design procedure step by step; the validity of the procedure is verified using full-wave EM simulations and experimental characterization of a fabricated prototype of the proposed device. Experimental characterizations of this device show that it has a stable frequency response as a function of angle of incidence for both the transverse electric (TE) and the transverse magnetic (TM) polarizations. This stability is attributed to the extremely small overall thickness of the structure as well as its small unit cell dimensions.