Disclosed is a cylindrical elastic mount including an inner sleeve member having an outward flange portion at one and an outer sleeve member disposed around the inner sleeve member with a radial spacing therebetween. The outer sleeve member has an outward flange portion at one end on the side of the flange portion of the inner sleeve member so that the flange portions of the inner and outer sleeve members are opposed to each other with an axial spacing between respective opposite surfaces of the flange portions. An elastic body disposed in the radial spacing between the inner and outer sleeves and the axial spacing between the flange portions for elastically connecting the inner and outer sleeve member. The opposite surfaces of the flange portions of the inner and outer sleeve members have respective tapered surfaces, which extend approximately parallel to each other in an axially outward direction inclined to a central axis of the mount, and are elastically connected with each other via the elastic body interposed therebetween.