In　this　paper,　we　introduce　the　concept　of　weak　Gabor　bi-frame　(WGBF)　in　a　general　closed　subspace　M　of　L-2(R).　It　is　a　generalization　of　Gabor　bi-frame,　and　is　new　even　if　M　=　L-2(R).　A　WGBF　for　M　contains　all　information　of　M　to　some　extent.　Let　a,　b　＞　0,　and　S　be　an　aZ-periodic　subset　of　R　with　positive　measure.　This　paper　is　devoted　to　characterizing　WGBFs　for　L-2(S)　of　the　form　G(g,　a,　b)　=　{e(2　pi　mbx)g(x　-　na)　:　m,　n　is　an　element　of　Z}.　It　is　well-known　that,　if　S　not　equal　R,　the　projections　of　Gabor　frames　for　L-2(R)　onto　L-2(S)　cannot　cover　all　Gabor　frames　for　L-2(S).　This　paper　presents　a　Zak　transform-domain　and　a　time-domain　characterization　of　WGBFs　for　L-2(S).　These　characterizations　are　new　even　if　S　=　R.　Some　examples　are　also　provided　to　illustrate　the　generality　of　our　theory.