A family of the diamond-type hierarchical lattices as a kind of fractals is proposed, on which the Ising model is exactly solved. The convergent condition of the free energy per bond in the thermodynamic limit is obviously given. We find that the unstable fixed point of the renormalization-group transformation moves toward K=0 (T=\ensuremath{\infty}) from K=\ensuremath{\infty} (T=0) as the number of branches P is increased, and when P=\ensuremath{\infty} (i.e., ${d}_{f}$=\ensuremath{\infty}) the unstable fixed point K=0 exactly, in contrast with that of the Bethe lattice. We also calculate the critical exponent of the correlation length; we find that in the ${d}_{f}$=2 and 3 cases the exponent is different from that of the regular lattice with d=2 and 3, respectively, which seems to imply that more general criteria for the classification of universality should be proposed. We have also discussed the upper critical fractal dimension.