This paper proposes a dual-band reconfigurable antenna array (DBRAA), enabling wireless capabilities in both sub-6 GHz (sub-6G) and millimeter wave (mmWave) bands using a single array. For the sub-6G band, we propose a reconfigurable antenna selection structure, where each sub-6G antenna is formed by multiplexing several mmWave antennas, with its position dynamically adjusted using PIN diodes. For the mmWave band, we develop a reconfigurable hybrid beamforming structure that connects radio frequency chains to the antennas via phase shifters and a reconfigurable switch network. We then investigate integrated sensing and communications (ISAC) in sub-6G and mmWave bands using the proposed DBRAA and formulate a dual-band ISAC beamforming design problem. This problem aims at maximizing the mmWave communication sum-rate subject to the constraints of sub-6G communication quality of service and sensing beamforming gain requirements. The dual-band ISAC beamforming design is decoupled into sub-6G beamforming design and mmWave beamforming design. For the sub-6G beamforming design, we develop a fast search-based joint beamforming and antenna selection algorithm. For the mmWave beamforming design, we develop an alternating direction method of multipliers-based reconfigurable hybrid beamforming algorithm. Simulation results demonstrate the effectiveness of the proposed methods.