We study the approximation of solutions to the stationary Stokes problem with a piecewise constant viscosity coefficient (interface Stokes problem) in the discrete duality finite volume (DDFV) framework. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first show how to design our new scheme, called m-DDFV, and we analyze the well-posedness of the scheme and its convergence properties. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for Stokes problems with discontinuous viscosity.