We evaluate a quasi-spherical, copper, microwave cavity resonator for accurately measuring the relative dielectric permittivity epsilon(r)(p,T) of helium and argon. In a simple, crude approximation the cavity's shape is a triaxial ellipsoid with axes of length a,1.001a and 1.005a, with a=5 cm. The unequal axes of the quasi-sphere separated each of the triply degenerate microwave resonance frequencies of a sphere (f(11)(TM),f(12)(TM),...,f(11)(TE),f(12)(TE),...) into three nonoverlapping, easily measured, frequencies. The frequency splittings are consistent with the cavity's shape, as determined from dimensional measurements. We deduced epsilon(r)(p,T) of helium and of argon at 289 K and up to 7 MPa from the resonance frequencies f(ln)(sigma), the resonance half-widths g(ln)(sigma), and the compressibility of copper. Simultaneous measurements of epsilon(r)(p,T) with the quasi-spherical resonator and a cross capacitor agreed within 1x10(-6) for helium, and for argon they differed by an average of only 1.4x10(-6). This small difference is within the stated uncertainty of the capacitance measurements. For helium, the resonator results for epsilon(r)(p,T) were reproducible over intervals of days with a standard uncertainty of 0.2x10(-6), consistent with a temperature irreproducibility of 5 mK. We demonstrate that several properties of quasi-spherical cavity resonators make them well suited to epsilon(r)(p,T) determinations. Ultimately, a quasi-spherical resonator may improve dielectric constant gas thermometry and realize a proposed pressure standard based on epsilon(r)(p,T).