Despite that the spline theory is a well studied topic, its relationship with the fractal theory is novel. Fractal approach offers a single specification for a large class of interpolants of which the classical spline is a particular member, and hence possesses considerable flexibility in the choice of an interpolant. The explicit construction of a C1-cubic Hermite fractal interpolation function (FIF) is introduced in the present work. If slopes at knot points are not known, then they are calculated through solution of a suitable linear system of equations so as to have C2 global smoothness for the resulting cubic FIF. Thus, the present method generalizes the classical C1-cubic Hermite and C2-cubic spline interpolants simultaneously, and offers a new approach to the development of cubic spline FIF in contrast to the construction through moments by Chand and Kapoor [SIAM J. Numer. Anal., 44(2), (2006), pp. 655-676]. It is shown that, for appropriate values of vertical scaling factors involved in the definition, developed C1-cubic Hermite FIF converges uniformly to the data generating function Φ ∈ C4 at least as rapidly as fourth power of the mesh norm approaches zero.