We report a novel hemispherical micro-cavity that is comprised of a planar integrated semiconductor distributed Bragg reflector (DBR) mirror, and an external, concave micro-mirror having a radius of curvature 50 microm. The integrated DBR mirror containing quantum dots (QD), is designed to locate the QDs at an antinode of the field in order to maximize the interaction between the QD and cavity. The concave micro-mirror, with high-reflectivity over a large solid-angle, creates a diffraction-limited (sub-micron) mode-waist at the planar mirror, leading to a large coupling constant between the cavity mode and QD. The half-monolithic design gives more spatial and spectral tuning abilities, relatively to fully monolithic structures. This unique micro-cavity design will potentially enable us to both reach the cavity quantum electrodynamics (QED) strong coupling regime and realize the deterministic generation of single photons on demand.