The numerical solution of the Stokes equation with discontinuous viscosity and singular force term is challenging, due to the discontinuity of pressure, non-smoothness of velocity, and coupled discontinuities along interface.In this paper, we give an efficient algorithm to solve this problem by employing spectral Legendre and Chebyshev approximations.First, we present the algorithm for a problem defined in rectangular domain with straight line interface. Then it is generalized to a domain with smooth curve boundary and interface by employing spectral element method. Numerical experiments demonstrate the accuracy and efficiency of our algorithm and its spectral convergence.