We have developed a millimeter-wave electron-spin-resonance (ESR) system for very low temperatures (T < 1 K) that can be employed for nuclear-magnetic-resonance measurements by using dynamic nuclear polarization. The system uses a Fabry-Perot resonator that works in the frequency range of 125 - 130 GHz and covers the temperature range of 0.09 - 6.5 K. We have performed ESR measurements in the frequency around 128 GHz by using MnxMg1-xO (x = 1.0 x 10(-4)) and free-radical samples of 1, 1-diphenyl-2-picrylhydrazyl (DPPH), because these samples have been proposed as field and sensitivity markers. Temperature dependence of the ESR signal intensity for MnxMg1-xO shows anomalies originating from magnetic order are found around 3.5 - 4 K. We estimate the sensitivity of the system for ESR detections to be 6 x 10(13) spins/G at 5.8 K. Because DPPH shows no observable shift in the magnetic field, we propose it as a useful standard marker for ESR measurements at very low temperatures.