A novel dual-band reconfigurable intelligent surface (DBI-RIS) design that combines the functionalities of millimeter-wave (mmWave) and sub-6 GHz bands within a single aperture is proposed. This design aims to bridge the gap between current single-band reconfigurable intelligent surfaces (RISs) and wireless systems utilizing sub-6 GHz and mmWave bands that require RIS with independently reconfigurable dual-band operation. The mmWave element is realized by a double-layer patch antenna loaded with 1-bit phase shifters, providing two reconfigurable states. An 8x8 mmWave element array is selectively interconnected using three RF switches to form a reconfigurable sub-6 GHz element at 3.5 GHz. A suspended electromagnetic band gap (EBG) structure is proposed to suppress surface waves and ensure sufficient geometric space for the phase shifter and control networks in the mmWave element. A low-cost planar spiral inductor (PSI) is carefully optimized to connect mmWave elements, enabling the sub-6 GHz function without affecting mmWave operation. Finally, prototypes of the DBI-RIS are fabricated, and experimental verification is conducted using two separate measurement testbeds. The fabricated sub-6 GHz RIS successfully achieves beam steering within the range of -35 to 35 degrees for DBI-RIS with 4x4 sub-6 GHz elements, while the mmWave RIS demonstrates beam steering between -30 to 30 degrees for DBI-RIS with 8x8 mmWave elements, and have good agreement with simulation results.