We have studied the signal received by a distant observer from a pulsar orbiting a stellar black hole. The binary system has a non-eccentric orbit and the most favourable configuration (edge-on) for observing the influence of the black hole on the signal emitted by the pulsar. In particular, the influence of the spin of the black hole on the observed signal and the possibility of determining the spin are examined in detail. In the case of quasi-coalescent binary systems, with an initial separation of ~0.01 light-seconds, the flux-time relation is different for a non-rotating black hole from that for a black hole with an extreme rotation velocity. The observational data obtained during the short lifetime of a quasi-coalescent system (<3 h) could be matched with realistic theoretical profiles, which will be sensitive to the specific spin of the black hole. In the case of binary systems with an initial separation of ~10 light-seconds, the probability of measuring the specific spin is briefly discussed on the basis of the signal travelling in the surroundings of the black hole, which is strongly affected by the rotation.