An improved tapping sleeve assembly comprising a sleeve configured to clamp a main fluid conduit oriented about a longitudinal axis, the sleeve having a connecting assembly between a first member and a second member, the connection assembly configured and arranged to tighten the first and the second members to the main fluid conduit, a gasket disposed between the first member and the main fluid conduit, the gasket comprising a flexible sheet having an inwardly facing surface and an outwardly facing surface and a thickened aperture seal configured to extend around a branch aperture in the first member between the sleeve and the main fluid conduit, and the aperture seal having a non-uniform radial thickness around the branch aperture when the connection assembly is in a non-actuated position.