We consider the axisymmetric flow of a Newtonian fluid associated with the spreading of a thin liquid film on a rotating annular disk. The effects of surface tension and gravity terms are included. The asymptotic solution for the free surface of the thin film is found using an expansion for the film thickness in powers of a small parameter characterizing the thickness of the film and applying the method of matched asymptotic expansion. This solution can be used to calculate the thickness of the film, the velocity field, and the pressure at any point on the disk with good accuracy. Numerical results are presented for a specific initial distribution of the film thickness. Many features of the spin-coating thinning process are captured by our asymptotic solution. We also produce results which are in excellent agreement with the experimental findings of Daughton and Givens (6) and Hwang and Ma (9).