In　this　paper,　we　consider　solutions　to　the　incompressible　axisymmetric　Euler　equations　without　swirl.　The　main　result　is　to　prove　the　global　existence　of　weak　solutions　if　the　initial　vorticity　w(0)(theta　)　satisfies　that　w(0)(theta/)r　is　an　element　of　L-1　boolean　AND　L-p(R-3)　for　some　p　＞　1.　It　is　not　required　that　the　initial　energy　is　finite,　that　is,　the　initial　velocity　u(0)　belongs　to　L-2(R-3)　here.　We　construct　the　approximate　solutions　by　regularizing　the　initial　data　and　show　that　the　concentrations　of　energy　do　not　occur　in　this　case.　The　key　ingredient　in　the　proof　lies　in　establishing　the　L-loc(2+alpha)(R-3)　estimates　of　velocity　fields　for　some　alpha　＞　0,　which　is　new　to　the　best　of　our　knowledge.