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In a mesoscopic conductor, electric resistance is detected even if the device is defect-free. We engineer and study a cold-atom analogue of a mesoscopic conductor. It consists of a narrow channel connecting two macroscopic reservoirs of fermions that can be switched from ballistic to diffusive. We induce a current through the channel and find ohmic conduction, even when the channel is ballistic. We measure in situ the density variations due to the presence of a current, and observe that density remains uniform and constant inside the ballistic channel. In contrast, for the diffusive case with disorder, we observe a density gradient extending through the channel. Our approach opens the way towards quantum simulation of mesoscopic devices with quantum gases.