We propose a new microwave resonator technique of permittivity measurement in a small volume of a high lossy liquid sample. The technique is based on the use of a metal semispherical resonator filled with dielectric. The resonator contains a small semispherical central cavity filled up with a monitored liquid. The characteristic equations for eigen frequencies of electromagnetic oscillations in this resonator has been obtained rigorously and analyzed owing to high simmetry of the resonator with the liquid. On this basis the operating mode of oscillations and optimal sizes of the measuring cavity and the resonator have been found. We discovered the effect of electromagnetic field extrusion out of the small central cavity filled up with the liquid that has big values (much more than one) of the real and imaginary parts of permittivity. The technique has been implemented experimentally in X band. The found theoretically effect of the field extrusion has been confirmed experimentally. The relations for the complex permittivity of liquid using the measured values of the resonator quality factor and resonant frequency has been found.