An　antimagic　labelling　of　a　graph　G　with　m　edges　is　a　bijection　f　:　E(G)　-＞　{1,　.　.　.　,　m}　such　that　for　any　two　distinct　vertices　u,　v　we　hav　Sigma(e　is　an　element　of　E(v))　f　(e)　not　equal　Sigma(e　is　an　element　of　E(u))　f　(e),　where　E(v)　denotes　the　set　of　edges　incident　v.　The　well-known　Antimagic　Labelling　Conjecture　formulated　in　1994　by　Hartsfield　and　Ringel　states　that　any　connected　graph　different　from　K-2　admits　an　antimagic　labelling.　A　weaker　local　version　which　we　call　the　Local　Antimagic　Labelling　Conjecture　says　that　if　G　is　a　graph　distinct　from　K-2,　then　there　exists　a　bijection　f　:　E(G)　-＞　{1,　.　.　.　,　vertical　bar　E(G)vertical　bar}　such　that　for　any　two　neighbours　u,　v　we　hav　Sigma(e　is　an　element　of　E(v))　f　(e)　not　equal　Sigma(e　is　an　element　of　E(u))　f　(e).　This　paper　proves　the　following　more　general　list　version　of　the　local　antimagic　labelling　conjecture:　Let　G　be　a　connected　graph　with　m　edges　which　is　not　a　star.　For　any　list　L　of　m　distinct　real　numbers,　there　is　a　bijection　f　:　E(G)　-＞　L　such　that　for　any　pair　of　neighbours　u,　v　we　have　that　Sigma(e　is　an　element　of　E(v))　f　(e)　not　equal　Sigma(e　is　an　element　of　E(u))　f　(e).