This　paper　presents　error　analysis　of　a　stabilizer　free　weak　Galerkin　finite　element　method　(SFWG-FEM)　for　second　-order　elliptic　equations　with　low　regularity　solutions.　The　standard　error　analysis　of　SFWG-FEM　requires　additional　regularity　on　solutions,　such　as　H-2　-regularity　for　the　second　-order　convergence.　However,　if　the　solutions　are　in　H(1+s　)with　0　＜　s　＜　1,　numerical　experiments　show　that　the　SFWG-FEM　is　also　effective　and　stable　with　the　(1　+　s)　-order　convergence　rate,　so　we　develop　a　theoretical　analysis　for　it.　We　introduce　a　standard　H-2　finite　element　approximation　for　the　elliptic　problem,　and　then　we　apply　the　SFWG-FEM　to　approach　this　smooth　approximating　finite　element　solution.　Finally,　we　establish　the　error　analysis　for　SFWG-FEM　with　low　regularity　in　both　discrete　H-1　-norm　and　standard　L-2　-norm.　The　(P-k(T),　Pk-1(e),　[Pk+1(T)](d))　elements　with　dimensions　of　space　d　=　2,　3　are　employed　and　the　numerical　examples　are　tested　to　confirm　the　theory.