A superconducting 1.3 GHz RF cavity is designed for acceleration of H − ions traveling at 81% of the speed of light. This 1.3 GHz cavity is considered for use in the squeezed International Linear Collider (ILC) section of the proposed high intensity H − ion linac at Fermilab, in order to accelerate a H − ion beam for the energy range of 420 MeV-1.2 GeV. The design of the earlier proposed 11-cell elliptical cavity is studied for higher order modes and the shape of the end cells is optimized to avoid trapped modes while keeping the field flatness and operating frequency undisturbed. A new alternative design based on nine cells is also proposed. Both inner and end cells are designed to achieve the optimal performance of the proposed cavity and higher order modes are studied.