The　well-known　density　theorem　for　one-dimensional　Gabor　systems　of　the　form　{e(2　pi　imbx)　g(x　-　na)}(m,n　is　an　element　of　Z),　where　g　is　an　element　of　L-2(R),　states　that　a　necessary　and　sufficient　condition　for　the　existence　of　such　a　system　whose　linear　span　is　dense　in　L-2(R),　or　which　forms　a　frame　for　L-2(R),　is　that　the　density　condition　a　b　＜=　1　is　satisfied.　The　main　goal　of　this　paper　is　to　study　the　analogous　problem　for　Gabor　systems　for　which　the　window　function　g　vanishes　outside　a　periodic　set　S　subset　of　R　which　is　a　Z-shift　invariant.　We　obtain　measure-theoretic　conditions　that　are　necessary　and　sufficient,　for　the　existence　of　a　window　g　Such　that　the　linear　span　of　the　corresponding　Gabor　system　is　dense　in　L-2(S).　Moreover,　we　show　that　if　this　density　condition　holds,　there　exists,　in　fact,　a　measurable　set　E　subset　of　R　with　the　property　that　the　Gabor　system　associated　with　the　same　parameters　a,　b　and　the　window　g　=　chi　E,　forms　a　tight　frame　for　L-2(S).　(C)　2008　Elsevier　Inc.　All　rights　reserved.