A new C1-rational cubic fractal interpolation surface is introduced to interpolate the surface data which lies on a rectangular grid. At the beginning, C1-rational cubic fractal interpolation functions are constructed along the grid lines. Then, the C1-rational cubic fractal interpolation surface is constructed with the help of blending functions and the rational cubic fractal interpolation functions. Convergence analysis of the fractal interpolation surface to an original function is carried out. Further, the scaling factors and the shape parameters are constrained, so that the fractal interpolation surface would lie above the fixed plane whenever the surface data lie above the plane. Also, conditions on the scaling factors and the shape parameters are derived to interpolate the convex surface data. © 2018, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.