A Mott insulator is a material that is insulating because of strong Coulomb repulsions between electrons. Doping charge carriers, electrons or holes into a Mott insulator can induce high-temperature superconductivity. Thus, what exactly happens when a charge carrier is doped into a Mott insulator is a key question in many-body physics 1-4 . To address this issue, ideally one should start from a zero-doping state and be able to introduce both holes and electrons in the dilute limit. However, such an idealized experiment has been impossible because of the lack of suitable materials. Here we show that a new 'ambipolar' cuprate makes it possible for the first time to cross the zero-doping state in the same material, which in turn allows us to address the physics of the extremely low-doping region. Surprisingly, we found that the antiferromagnetic ground state sharply changes between electron- and hole-doped sides, and this change is dictated by the existence of only 0.1 ppm of charge carriers. Moreover, we observed that the Néel temperature T N shows an unexpected reduction in a narrow range centred at the zero-doping state, across which the system exhibits asymmetric behaviours in transport measurements. Our findings reveal the inherently different nature of electron and hole doping in the dilute limit of a Mott-insulating cuprate.