ABSTRACT The analysis of the thermal spectrum of geometrically thin and optically thick accretion discs of black holes, the so-called continuum-fitting method, is one of the leading techniques for measuring black hole spins. Current models normally approximate the disc as infinitesimally thin, while in reality the disc thickness is finite and increases as the black hole mass accretion rate increases. Here we present an XSPEC model to calculate the multitemperature blackbody spectrum of a thin accretion disc of finite thickness around a Kerr black hole. We test our new model with an RXTE observation of the black hole binary GRS 1915+105. We find that the spin value inferred with the new model is slightly higher than the spin value obtained with a model with an infinitesimally thin disc, but the difference is small and the effect is currently subdominant with respect to other sources of uncertainties in the final spin measurement.