A nonlinear optical plasmonic core-shell nanocavity is demonstrated as an efficient, subwavelength coherent light source through second-harmonic generation. The nonlinear optical plasmonic nanocavity incorporates a noncentrosymmetric medium, which utilizes the entire mode volume for even-order nonlinear optical processes. In previous plasmonic nanocavities, enhancement of such processes was only possible at the interface but symmetry prohibited in the body. We measured an enhancement of over 500 times in the second-harmonic radiation power. Calculations show that an enhancement of over 3500 times is achievable.