Traditionally, electrical noise is considered as an interference source for low level measurements. Shot noise is the current fluctuation caused by the discreteness of electrons. In a mesoscopic system, shot noise is sensitive to the interaction of charge carriers. Since the 20^th century, it has been found that the shot noise measurement can provide the information about quantum fluctuations, which cannot be measured with traditional transport measurement method. It is usually difficult to measure weak noise signal at ultra- low temperature due to technical difficulties. It is necessary to mount a cryogenic preamplifier close to the sample to improve signal-to-noise ratio and to increase the bandwidth. Therefore, the ultra-low background noise and the power consumption of the amplifier should be used. In this report we present a shot noise measurement system at dilution refrigerator temperatures. We also introduce and analyze the noise model of our shot noise measurement system. With customized high electron mobility transistors, we make a series of ultra-low noise cryogenic preamplifiers. All the electronic components of the amplifier are packed into a shielding box, which makes the installation of the cryogenic amplifier more convenient. The amplifier is mounted on the 4 K stage of a dry dilution refrigerator and the total power consumption is less than 0.754 mW. The gains and the background noises of the amplifiers are calibrated with the Johnson-Nyquist noise of the combination of a superconducting resistor and a normal resistor at various temperatures. The measured input referred noise voltage can be as low as 0.25 nV/√Hz. In this report, the performance of the system is demonstrated by the shot noise measurement of an Al/AlO_x/Al tunnel junction at various temperatures. Above the superconducting transition temperature of aluminum, the measured Fano factor of the system is very close to 1, which is in a good agreement with the theory prediction.