A damper for damping the vibration of a building caused by an earthquake, comprising: at least a hollow outer elongated body and a hollow inner elongated body, said inner elongated body including an inserted portion which has an outer surface extending in an axial direction and disposed with a gap with respect to an axially extending inner surface of said outer elongated body, and which is inserted in said outer elongated body so as to be relatively movable in the axial direction, a viscoelastic body, which is set in a hermetically sealed state, being disposed in the gap between the inner surface of said outer elongated body and the outer surface of said inner elongated body in such a manner as to be in contact with the inner surface and the outer surface, a one-side attaching plate member being secured to one axial end portion of said outer elongated body, and an air chamber whose volume is capable of decreasing or increasing, one end portion of said inner elongated body integrally extending from said inserted portion in the axial direction and projecting to the outside from another axial end portion of said outer elongated body, an other-side attaching plate member is secured to the one end portion of said inner elongated body, said air chamber is formed in an interior of said inner elongated body and the volume of said air chamber decreases or increases due to the increase or decrease of the internal pressure of the viscoelastic body disposed in the interior of said inner elongated body, based on the extension or retraction of said inserted portion of said inner elongated body in the axial direction with respect to an interior of said outer elongated body. at least a hollow outer elongated body and a hollow inner elongated body, said inner elongated body including an inserted portion which has an outer surface extending in an axial direction and disposed with a gap with respect to an axially extending inner surface of said outer elongated body, and which is inserted in said outer elongated body so as to be relatively movable in the axial direction, a viscoelastic body, which is set in a hermetically sealed state, being disposed in the gap between the inner surface of said outer elongated body and the outer surface of said inner elongated body in such a manner as to be in contact with the inner surface and the outer surface, a one-side attaching plate member being secured to one axial end portion of said outer elongated body, and an air chamber whose volume is capable of decreasing or increasing, one end portion of said inner elongated body integrally extending from said inserted portion in the axial direction and projecting to the outside from another axial end portion of said outer elongated body, an other-side attaching plate member is secured to the one end portion of said inner elongated body, said air chamber is formed in an interior of said inner elongated body and the volume of said air chamber decreases or increases due to the increase or decrease of the internal pressure of the viscoelastic body disposed in the interior of said inner elongated body, based on the extension or retraction of said inserted portion of said inner elongated body in the axial direction with respect to an interior of said outer elongated body.