An advanced undergraduate laboratory experiment is outlined which uses a dye laser to map out the chromatic dispersion curve of a polymethyl methacrylate (PMMA) optical fibre. Seven different wavelengths across the visible spectrum are employed using five different dyes. The light pulse is split into two pulses, one to a nearby photodetector and the other coupled to the optical fibre cable at the end of which there is another photodetector. The difference in time of arrival at the detectors is used to compute the speed of light in the fibre for a given colour. In addition to a pedagogically simple and intuitive demonstration of the measurement of index of refraction, the use of a long fibre eliminates the need to direct the dangerous UV/visible laser pulse beam across a classroom, as is usually done in similar experiments. Ways to avoid systematic errors and other technical pitfalls--such as ringing oscillations--are presented. (Contains 8 figures.)