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We present direct measurements of the hyperfine splitting of Rydberg statesin rubidium 87 using Electromagnetically Induced Transparency (EIT)spectroscopy in a room-temperature vapour cell. With this method, and in spiteof Doppler-broadening, line-widths of 3.7 MHz FWHM, i.e. significantly belowthe intermediate state natural linewidth are reached. This allows resolvinghyperfine splittings for Rydberg s-states with n=20...24. With this method weare able to determine Rydberg state hyperfine splittings with an accuracy ofapproximately 100 kHz. Ultimately our method allows accuracies of order 5 kHzto be reached. Furthermore we present a direct measurement ofhyperfine-resolved Rydberg state Stark-shifts. These results will be of greatvalue for future experiments relying on excellent knowledge of Rydberg-stateenergies and

High-Precision Measurement of Rydberg State Hyperfine Splitting in a Room-Temperature Vapour Cell