We present a new and highly stable cylindrical microwave cavity resonator designed for the measurement of the relative permittivities of gases. The cylinder has an internal volume of just 5.7 cm(3), which makes it the smallest such a resonator reported to date. The equipment and procedure used to measure the resonance frequencies and halfwidths with fractional random errors better than +/- 6 . 10(-7) and +/- 4 . 10(-3), respectively, are described. We have developed also an analytical model which describes the cylinder's resonant behaviour with a fractional accuracy of the order of +/- 10(-6). Our model includes a novel method for determining the perturbation due to the presence of an opening in the boundary wall which is shown to be in excellent agreement with experimental results.