We have made direct measurements of the thermal boundary resistance, R(bd), between high quality epitaxial films of YBa2Cu3O7-delta and a variety of substrates with and without buffer layers. The boundary resistance was deduced from measurements of the electrical resistance changes in three parallel strips of YBa2Cu3O7-delta when one was electrically heated. Our measurements indicate that R(bd) is weakly dependent on temperature and, for all the measured samples, has a value of 0.8-1.4 x 10(-3) K cm2/W between 90 and 200 K, which is a factor of almost-equal-to 80 larger than the prediction of the acoustic mismatch model at 100 K. The thermal response time of almost-equal-to 1 ns which results from the heat capacity of the film and R(bd) is observed in many experiments with pulsed laser sources.