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The key to being able to operate the superconducting solenoids in the Muon Ionization Cooling Experiment (MICE) using cryocoolers running at around 4.2 K is the application of high temperature superconducting (HTS) leads. Because the MICE magnets are not shielded, all of them will have a stray magnetic field in the region where the coolers and the HTS leads are located. The behavior of the HTS leads depends strongly on the HTS material used for the leads, the magnetic field and their warm end temperature. A pair of binary leads consisting of copper leads and HTS leads made from oriented multiple strands of BSCCO wires will be used for electrical transfer of the MICE coupling magnet for the purpose of reducing the heat leak through the leads to 4.2 K region. This paper mainly discusses the detailed design of the HTS leads and their cooling. Protection for the HTS leads during a power failure is discussed as well.