We propose a new way to hide the fifth dimension, and to modify gravity in the far infra-red. A gravitating tensional membrane in five dimensions folds the transverse space into a truncated cone, stoppered by the membrane. For near-critical tension, the conical opening is tiny, and the space becomes a very narrow conical sliver. A very long section, of length comparable to the membrane radius divided by the remaining conical angle, of this sliver is well approximated by a narrow cylinder ending on the membrane. Inside this cylindrical throat we can reduce the theory on the circle. At distances between the circle radius and the length of the cylinder, the theory looks 4D, with a Brans-Dicke-like gravity, and a preferred direction, while at larger distances the cone opens up and the theory turns 5D. The gravitational light scalar in the throat can get an effective local mass term from the interplay of matter interactions and quantum effective potentials on the cone, which may suppress its long range effects. We discuss some phenomenologically interesting consequences.