The solute rich liquid-crystal pulling method, a modified Czochralski-pulling technique, can be used to grow very large single crystals, larger than 1 cm along each axis. In order to examine the crystallinity of the YBa2Cu3Oy single crystals grown by this technique, neutron diffraction was applied for six crystals subjected to various annealing conditions. The typical mosaicity was smaller than 1 degrees along the c-axis and 2 degrees in the ab-plane. Some crystals were of high quality, with a mosaicity of less than 0.2 degrees for every direction. The oxygen inhomogeneity present in the as-grown samples was eliminated by prolonged-duration oxygen annealing. Furthermore, we succeeded in observing the orthorhombic II and III (ortho-II and III) phases, and determined the correlation lengths along the a-axis as xi(a) = 16 Angstrom and xi(a) = 19 Angstrom for y approximate to 6.60 (ortho-II) and y approximate to 6.70 (ortho-III), respectively.