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A new fractional operator with Rabotnov fractional exponential kernel was recently introduced in the literature. In this paper, we consider the fractional diffusion equation in the context of this new fractional operator. We determine the form of the analytical solution and the displacement of the fractional diffusion equation generated by this new fractional operator. In other words, this paper is an application of the fractional derivative with Rabotnow exponential kernel into the diffusion processes. We discuss the results of this paper physically. We make some comparison studies between the results generated by this new operator and those obtained with the fractional derivatives with the singular kernel. We illustrate the main findings of this work by graphical representations of the obtained solution.