The photonic-band-gap (PBG) structure composed of an anisotropic-dielectric sphere in uniform dielectric medium is studied by solving Maxwell's equations using the plane-wave expansion method. In particular, for a uniaxial material with large principal refractive indices and sufficient anisotropy between them, the photonic band structures possess a full band gap in the whole Brillouin zone for a diamond lattice. Furthermore, in the 1/3 partial Brillouin zone where the Bloch wave vector has a dominant component along the extraordinary axis of uniaxial sphere, the photonic band structures are found to exhibit full band gaps for all the other lattices such as face-centered-cubic, body-centered-cubic, and simple-cubic lattices, although a complete band gap does not open in the whole Brillouin zone. The partial band gaps persist at a very low filling fraction of uniaxial sphere. This phenomenon is attributed to the breakdown of the photonic band degeneracy at high-symmetry points of the Brillouin zone by the anisotropy of material dielectricity. The combination of such an anisotropic PBG structure with the self-arrangement technique of colloidal crystal may provide a possible way to fabricate the three-dimensional photonic crystal in visible and infrared regimes. The application of a strong electric field may bring into alignment the extraordinary axis of uniaxial sphere as this configuration of spheres is most favorable thermodynamically.