We present a general relativistic accretion disc model and its application to the soft-state X-ray spectra of black hole binaries. The model assumes a flat, optically thick disc around a rotating Kerr black hole. The disc locally radiates away the dissipated energy as a blackbody. Special and general relativistic effects influencing photons emitted by the disc are taken into account. The emerging spectrum, as seen by a distant observer, is parametrized by the black hole mass and spin, the accretion rate, the disc inclination angle and the inner disc radius. We fit the ASCA soft-state X-ray spectra of LMC X-1 and GRO J1655-40 by this model. We find that, having additional limits on the black hole mass and inclination angle from optical/UV observations, we can constrain the black hole spin from X-ray data. In LMC X-1 the constraint is weak, and we can only rule out the maximally rotating black hole. In GRO J1655-40 we can limit the spin much better, and we find 0.68 ⩽ a ⩽ 0.88. Accretion discs in both sources are radiation-pressure dominated. We do not find Compton reflection features in the spectra of any of these objects.