High-purity and homogeneous YBa2Cu3Oy single crystals with carrier dopinglevel near the AFM-SC boundary have been obtained in the oxygen content rangebetween y = 6.340 and 6.370. The crystals are ortho-II phase at roomtemperature and undergo the orthorhombic to tetragonal transition at about140_Degree_C. They show sharp superconducting transitions, with Tc between 4and 20 K. Tc changes by 0.8 K when the oxygen content y is changed by 0.001,and is also sensitive to annealing conditions near room temperature, due to thedependence of doping on oxygen ordering correlation lengths. Crystals withoxygen content y lower than 6.345 are non-superconducting.