The current carrying capabilities of high critical temperature (Tc) technical conductors have considerably improved in recent years, and this has practical consequences on the requirements of the measurement equipment. We have designed, constructed, and tested a probe for high-current, high-precision critical current measurements in liquid helium and in gas flow at different temperatures. To minimize the ohmic heating and the thermal conduction losses, the probe employs high-temperature superconductors (HTS) current leads rated at 2000 A, which are cooled by the exhaust helium gas at temperatures below 70 K. An active temperature stabilization system keeps the sample temperature constant at the target during current runs with a precision of ±0.01 K. It consists of a proportional-integral-differential loop that controls heaters close to the sample taking the current-induced heating effects into account. We present the design of the probe and data collected on HTS from different manufacturers, which highlight the performance of the system