The problem of measuring very low levels of current noise in bipoles (linear or not) is dealt with, and a new measurement technique is proposed, which allows us to measure noise power spectra 6-10 dB lower than the equivalent input power spectrum of the amplifier necessary to perform the measurement. Thus we obtain an improvement of 16-20 dB in the sensitivity with respect to the one of conventional methods, which, for an acceptable accuracy, require the noise of the bipole under test to be 10 dB larger than the equivalent input one of the amplifier. Our method is based upon the accurate measurement of the amplifier transimpedance with respect to the input current noise sources, and on the precise evaluation and subtraction of the contribution from all the spurious sources to the total noise. The whole procedure is implemented by means of a dual channel signal analyzer, and almost completely automatized. The experimental results obtained with the application of our method agree very well with theoretical previsions.