In this paper, we study the equation system governing the movement of an immersed interface in incompressible fluid flows, and propose an efficient method for its numerical solution. The particularity of the current model is the inextensibility constraint imposed on the interface. We are interested in constructing a suitable variational formulation associated to this problem and the well-posedness of the weak problem. The significance of this variational formulation is that both the inextensibility of the interface and fluid incompressibility are strictly satisfied, and the well-posedness of the associated weak problem is rigorously proved. To the best of the authors' knowledge, no other models can be claimed to posses these properties. In fact our new formulation renders the inextensibility and the incompressibility constraints into a unique saddle point problem. Then, based on the proposed variational framework, we design an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: 1) a variational framework for the weak solutions of the immersed interface/incompressible equations and rigorous proof of the well-posedness of the weak problem; 2) a spectral method for solving the weak problem, together with a detailed stability analysis for the numerical solutions; 3) efficient implementation technique for the proposed method and some numerical experiments carried out to confirm the theoretical claims.