A tubular vibration-damping mount comprising: a core member (1,11); a tubular member (2,12) located below the core member (1,11) and having an inside diameter larger than an outside diameter of the core member (1,11); a rubber member (3,13) located between the core member (1,11) and the tubular member (2,12) to couple the core member (1,11) and the tubular member (2,12) by vulcanization adhesion, an upper surface of the rubber member (3a,13a) having, as viewed in a cross section including an axis of the core member (1,11), a recessed portion in a radially inner direction of the rubber member (3b,13b) and an inclined portion whose inclination angle is smoothly and upwardly increased in the radially inner direction from a point (3c,13c) on the tubular member (2,12) as a starting point; and a bracket (4,14) supporting an outer periphery of the tubular member (2,12) with the tubular member (2,12) being inserted; wherein the tubular member (2,12) has, in an area higher than a lower end of an adhesion surface between the tubular member (2,12) and the rubber member (3,13), a straight cylindrical portion which has an inner periphery and an outer periphery, both of which are parallel to an axis of the tubular member (2,12) and a diameter-enlarged portion which has an inner periphery extending from an upper end of the straight cylindrical portion and smoothly enlarging the inside diameter of the tubular member (2,12) and the upper surface of the rubber member (3a,13a) starts from a point on the inner periphery of the diameter-enlarged portion of the tubular member (2,12), which the point agrees with the starting point, and wherein the outer periphery of the diameter-enlarged portion comprises a surface whose diameter is smoothly increased from the outer periphery of the straight cylindrical portion and the bracket (4,14) comprises a guide portion having an inside diameter larger than the largest outside diameter of the diameter-enlarged portion and is characterized in that a constant inside diameter portion located above the guide portion and having an inside diameter engaging the outside diameter of the straight cylindrical portion in a posture of supporting the tubular member (2,12). a core member (1,11); a tubular member (2,12) located below the core member (1,11) and having an inside diameter larger than an outside diameter of the core member (1,11); a rubber member (3,13) located between the core member (1,11) and the tubular member (2,12) to couple the core member (1,11) and the tubular member (2,12) by vulcanization adhesion, an upper surface of the rubber member (3a,13a) having, as viewed in a cross section including an axis of the core member (1,11), a recessed portion in a radially inner direction of the rubber member (3b,13b) and an inclined portion whose inclination angle is smoothly and upwardly increased in the radially inner direction from a point (3c,13c) on the tubular member (2,12) as a starting point; and a bracket (4,14) supporting an outer periphery of the tubular member (2,12) with the tubular member (2,12) being inserted; wherein the tubular member (2,12) has, in an area higher than a lower end of an adhesion surface between the tubular member (2,12) and the rubber member (3,13), a straight cylindrical portion which has an inner periphery and an outer periphery, both of which are parallel to an axis of the tubular member (2,12) and a diameter-enlarged portion which has an inner periphery extending from an upper end of the straight cylindrical portion and smoothly enlarging the inside diameter of the tubular member (2,12) and the upper surface of the rubber member (3a,13a) starts from a point on the inner periphery of the diameter-enlarged portion of the tubular member (2,12), which the point agrees with the starting point, and wherein the outer periphery of the diameter-enlarged portion comprises a surface whose diameter is smoothly increased from the outer periphery of the straight cylindrical portion and the bracket (4,14) comprises a guide portion having an inside diameter larger than the largest outside diameter of the diameter-enlarged portion and is characterized in that a constant inside diameter portion located above the guide portion and having an inside diameter engaging the outside diameter of the straight cylindrical portion in a posture of supporting the tubular member (2,12).