The object of this paper is to present the exact solution of the fractional Cattaneo equation for describing anomalous diffusion. The classical Cattaneo model has been generalised to the space-time fractional Cattaneo model. The method of the joint Laplace and Fourier transform is used in deriving the solution. The solutions of the fractional Cattaneo equation are obtained under integral and series forms in terms of the H-functions. Finally, the fractional order moments are also investigated.