We measure the current noise of several cryogenic cables in a pulse tube based dilution refrigerator at frequencies between about 1~mHz and 50~kHz. We show that vibration-induced noise can be efficiently suppressed by using vacuum-insulated cables between room temperature and the 2nd pulse tube stage. A noise peak below 4 fA at the 1.4~Hz operation frequency of the pulse tube, and a white noise density of 0.44 fA/sqrt{Hz} in the millihertz range are obtained.
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