With an AVR-based Arduino like the Uno, you can use the 1.1V internal ADC reference to get ~1 mV resolution (1024 steps from 0 to about 1100 mV).
The reference is stable but not accurately calibrated, and can range from about 1.000 to 1.200 V, so you may want to calibrate it.
You may need to use shielded probe cables to reduce external electrical noise.