We studied experimentally the electrodynamic characteristics of a hemispherical shielded dielectric cavity in the 8-mm wavelength range. The use of a slot transmission line is proposed to excite whispering gallery modes in such a cavity. It was found that the proposed excitation method is efficient and allows one to generate high-Q higher modes in a shielded dielectric cavity introducing no additional energy losses. It was shown experimentally that the achieved intrinsic Q-factor of the cavity under consideration can exceed the threshold values limited by the dissipative losses in its material. This effect is determined by the shift of the resonance field to the air gap between the metal and dielectric elements of the cavity. Agreement of the calculated and experimental data is demonstrated.