In recent precise experiments, such as optical measurements with interferometry, physical quantities sometimes need to be observed in a 'quiet' vacuum, where all of the instruments in the vacuum are well isolated from the vibration of the vacuum tank including the vacuum-pumping system and, sometimes, from the seismic noise of the earth. We have developed a new technique for vibration isolation between the inside and outside of a vacuum (e.g. an optical bench and a tank), which involves a double-balanced bellows (DBB) to compensate for the atmospheric pressure. Both the detailed structure and the DBB mechanism as well as the results of measurements of the vibration characteristics are presented, showing the validity of using the DBB, especially in gravitational-wave experiments.