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.