The invention relates to a composition based on an ethylene-vinyl alcohol copolymer comprising: 50 to 98% by weight of an EVOH copolymer 1 to 50% by weight of a polyethylene (A) 1 to 15% by weight of a polymer (B) resulting from the reaction (i) of a copolymer (B1) of ethylene and of an unsaturated monomer X grafted or copolymerized with (ii) a polyamide (B2). It also relates to a multilayer structure comprising a layer consisting of this composition. This structure is useful for making packaging. The composition of the invention also adheres well to PET, this property being useful in coinjection to make bioriented preforms.