We report a cross-correlation measurement system, based on a new approach, which can be used to measure shot noise in a mesoscopic conductor at milliKelvin temperatures. In contrast to other measurement systems in which high-speed low-noise voltage amplifiers are commonly used, our system employs homemade transimpedance amplifiers (TAs). The low input impedance of the TAs significantly reduces the crosstalk caused by unavoidable parasitic capacitance between wires. The TAs are designed to have a flat gain over a frequency band from 2 kHz to 1 MHz. Low-noise performance is attained by installing the TAs at a 4 K stage of a dilution refrigerator. Our system thus fulfills the technical requirements for cross-correlation measurements: low noise floor, high frequency band, and negligible crosstalk between two signal lines. Using our system, shot noise generated at a quantum point contact embedded in a quantum Hall system is measured. The good agreement between the obtained shot-noise data and theoretical predictions demonstrates the accuracy of the measurements.