The new concept of the "$4\pi$-Continuous-Mode-Target" will allow the simultaneous usage of a large angular acceptance detector with a continuously polarised target. For polarising a target the process of Dynamic Nuclear Polarisation (DNP) is used which requires low temperatures and a high homogeneous magnetic field within the target volume. A thin, superconducting magnet as a part of the new horizontal cryostat creates a longitudinal magnetic field of 2.5 T with a field homogeneity of $10^{-4}$. Since the internal superconducting magnet is operated with a high direct current there has to be two current leads for connecting the magnet terminals found in the low temperature region to the power supply located at room temperature. Due to Fourier's Law and joule heating a current lead may imply a large heat load on the magnet and the cryostat. As the magnet is operated near its critical parameters, it is important to reduce the heat flux to the superconducting wire to an absolute minimum. The geometry of the normal conducting part of these current leads has been adapted to the cryostat requirements and has been optimised to minimize the heat load. A FVM calculation has been additonally used to check the fulfillment of the requirements.