We have developed a nuclear magnetic resonance spectrometer with a series tuned input circuit for measurements on samples at millikelvin temperatures based on an integrated two-stage superconducting quantum interference device current sensor, with an energy sensitivity ε=26±1h when operated at 1.4K. To maximize the sensitivity, both the nuclear magnetic resonance spectrometer pickup coil and tuning capacitor need to be cooled, and the tank circuit parameters should be chosen to equalize the contributions from circulating current noise and voltage noise in the superconducting quantum interference device. A noise temperature TN=7±2mK was measured, at a frequency of 0.884MHz, with the circuit parameters close to optimum.