To provide an elasticity connection method and a connecting member that saves on space and exhibits high vibration damping ability.

In a method of connecting a first member and a second member resiliently, a cylindrical member having rubber-like elasticity is fitted to a cylindrical portion of a joint member fixed to the first member and having the cylindrical portion projecting from the first member. The cylindrical member portion is inserted into a female screw hole which is formed on the second member to be vibration-isolated and jointed to the first member, and whose inside diameter is greater than the outside diameter of the cylindrical member. A coil is screwed in between the outer circumference of the cylindrical member and the female screw. By doing this, the second member is connected to the first member resiliently through the coil in contact with the outer circumference of the cylindrical member and the cylindrical member. Also, by cutting off the thread of the screw from the axial groove, it is attained to reduce the rigidity of the thread, to deform the thread easily by vibration force in the direction of the bolt shaft, and to damp the vibration force applied to the bolt by frictional resistance due to slipping on the thread joint surface generated by the deformation.