An advanced Microwave Limb Sounder (MLS), now in concept development for a potential future mission, is a space-borne heterodyne instrument to measure pressure, temperature, and atmospheric constituents from thermal emission between 120 and 2400 GHz. Previous MLS instruments used pencil-beam antennas sized to resolve ~1 vertical scale height. Current atmospheric models need better horizontal resolution than orbit spacing provides. To meet these needs, a new antenna concept combines the wide scan range of the parabolic torus with unblocked offset Cassegrain optics. The resulting system is diffraction-limited in the vertical plane but extremely astigmatic, with beamwidths 0.13×2.5°. Nadir axis symmetry ensures that this Beam Aspect Ratio (BAR) is invariant over ±33 degrees of azimuth. The antenna can feed either an array of receivers or multiplexed low-noise receivers whose FOVs are swept by a small scanning mirror. We describe 3 stages of antenna design: First, using a paraxial-optics method, we choose conic profiles given vertical resolution orbit geometry, then develop the surfaces by nadir axis rotation, matching axisymmetric feeds to the BAR. A ray-trace program validates the design and generates alignment and deformation tolerances. Finally, a physical optics analysis verifies reflector surface currents and radiation patterns.