It should be a cylinder-like thing with a vertical-convex bit which would correspond to the vertical bit of the cylinder, minus two filled cones.
This is far from a rigorous treatment of the problem, but my reasoning goes something along the lines of this:
Orient everything so the cone is horizontal, and the axis it's being spun around is vertical. Consider the cylinder, centered at the origin, which spans the height of the cone (which is equal to the cone's diameter) and goes out far enough to be tangent to the cone's base. This cylinder slices into the cone, making a circle-y cross-section on the cylinder. For every point on the outer wall of the cylinder, some point on the circle-y cross-section will be at the same height as that point on the outer wall, so while the cone sweeps around, the circle-y cross section will hit that point, so every point on the outer wall of the cylinder will be touched by the cone. Using smaller, similar cylinders gives us a series of outer-cylinder-walls whose heights are linearly increasing with radius that get hit by the cone, which corresponds to a cylinder minus two cones.
Now the circumference of the cone's base is further from the origin than the cone's base's center, though, so the curved heel of the cone's circumference will sweep out a convex surface encasing the cylinder. I believe the surface will be a section of an ellipse, swept around the circumference of the cylinder.
I'm unfortunately a little too tired to do calculations right now, so I can't really give a more rigorous treatment than the hand-wavey vague terms I just did.
