Three-dimensional dielectric optical crystals with a high index show a complete photonic bandgap (PBG), blocking light propagation in all directions. We show that this bandgap can be used to trap light in low-index defect cavities, leading to strongly enhanced local fields. We compute the band structure and optimize the bandgap of an inverse 3D rod-connected diamond (RCD) structure, using the plane-wave expansion (PWE) method. Selecting a structure with wide bandgap parameters, we then add air defects at the center of one of the high-index rods of the crystal and study the resulting cavity modes by exciting them with a broadband dipole source, using the finite-difference time-domain (FDTD) method. Various defect shapes were studied and showed extremely small normalized mode volumes (Veff) with long cavity storage times (quality factor Q). For an air-filled spherical cavity of radius 0.1 unit-cell, a record small-cavity mode volume of Veff~2.2 × 10−3 cubic wavelengths was obtained with Q~3.5 × 106.