This paper describes a study of the melting point of gallium with the new NIM Johnson noise thermometer (JNT). The new thermometer adopts the structure of switching correlator and commutator with the reference resistor maintained at the triple point of water. The electronic system of the new thermometer is basically the same as the current JNT, but the preamplifiers have been improved slightly. This study demonstrates that examining the characteristics of the noise signals in the frequency domain is of critical importance in constructing an improved new thermometer, where a power spectral analysis is found to be critical in establishing appropriate grounding for the new thermometer. The new JNT is tested on measurements of the thermodynamic temperature of the melting point of gallium, which give the thermodynamic temperature of 302.9160 K, with an overall integration time of 190 h and a combined standard uncertainty of 9.4 mK. The uncertainty analysis indicates that a standard combined uncertainty of 3 mK could be achieved with the new thermometer over an integration period of 1750 h.