The 22.9kV/630A superconducting fault current limiter (SFCL) is developed by the KEPRI-LSIS collaboration group. This resistive SFCL uses three pairs of conduction-cooled current leads. When the SFCL system is in the fault mode. the current flows 20 times more than the steady state. Therefore. it is important that the current lead is designed to have the thermal stability in order to minimize the heat input of the cold-end. This paper presents the design and performance results of a pair of conduction-cooled brass current leads considering both cases that the SFCL system operates at the steady state and the fault current.