In order to preserve a low emittance beam along the ~11 km long main linacs of the International Linear Collider (ILC), precise monitoring and control of the beam orbit is mandatory. A resolution <; 1 μm is required for the beam position monitors (BPM), which are located inside the cryomodules, i.e. operated at cryogenic temperatures. By using the electromagnetic field simulation programs (CST Studio Suite), we have designed a high resolution cold cavity BPM, operating in the L-Band at a 1.3 GHz dipole-mode frequency. This enables the measurement of the beam position with a Project-X like beam structure, as well as with ILC beam parameters at the superconducting RF test accelerator in the Fermilab New Muon Lab (NML) building. The results of the design studies predicts a resolution potential of a few hundred nanometers with a signal decay time of 50 ns.