The influence of both the finite band width of the laser pump and the effect of the frequency shift of the molecular vibration in the electronic transition on the intensity correlation function gamma ^2,2( tau ) of the resonance fluorescence light from a single two-atom molecule are studied. In the case of small vibrational damping a double-exponential asymptotic behaviour of gamma ^2,2( tau ) is found, which is independent of the bandwidth of the exciting laser light.