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This paper presents a digital stochastic instrument for the measurement of the harmonics of mains voltages and currents. The instrument operation is based on stochastic analog-to-digital (A/D) conversion and accumulation, with a novel hardware structure tailored for harmonic measurements. A theoretical approach in determining measurement uncertainty is first presented, and the accuracy trends with varying signal-to-noise ratios (SNRs) are then analyzed. Using this theoretical analysis, the relative standard uncertainty of measurement is found to be 0.32% at -10-dB SNR and less than 0.01% at 20-dB SNR. The method and the predicted uncertainty for 50 harmonics are first validated by simulation. The proposed technique is then implemented into a newly developed prototype instrument that was coupled with a coreless current measurement transformer. Calibration shows high linearity of the coreless transformer and, hence, of the whole instrument in a wide range of primary signals. Experimental results, which measure low-level and highly distorted currents, confirm both the theory and simulation.