Method for producing a multilayer data medium by means of hot pressure lamination of a stack of a plurality of superposed layers comprising at least one layer made of at least one thermoplastic material, able to be marked under the effect of a laser radiation applied to at least one outer face of this data medium, in which: - at least one layer, called a covering layer (21, 22): ∘ transparent to a marking laser radiation, ∘ extending between said outer face and a layer, called a laser-marking sublayer, sensitive to this laser radiation, so as to allow the production of a marking in the marking sublayer under the effect of this laser radiation passing through each covering layer (21, 22), ∘ incorporating at least one marking with an optically variable effect arranged relative to the laser-marking sublayer such that said outer face has a portion with an optically variable effect optically superposed on at least part of the laser-marking sublayer and a portion devoid of marking with an optically variable effect optically superposed on at least part of the laser-marking sublayer, characterized in that: - said at least one covering layer (21, 22) further incorporates at least one marking agent sensitive to said laser radiation so as to allow the production of a marking in said at least one covering layer (21, 22) under the effect of this laser radiation, - said portion devoid of marking with an optically variable effect and said portion with an optically variable effect are produced with contents of at least one marking agent sensitive to said laser radiation, between said outer face of the support and said laser-marking sublayer, which are different, and - said portion devoid of marking with an optically variable effect is devoid of marking agent sensitive to said laser radiation, at least one marking agent sensitive to said laser radiation being incorporated into at least one covering layer (21, 22) only in said portion with an optically variable effect. - at least one layer, called a covering layer (21, 22): ∘ transparent to a marking laser radiation, ∘ extending between said outer face and a layer, called a laser-marking sublayer, sensitive to this laser radiation, so as to allow the production of a marking in the marking sublayer under the effect of this laser radiation passing through each covering layer (21, 22), ∘ incorporating at least one marking with an optically variable effect arranged relative to the laser-marking sublayer such that said outer face has a portion with an optically variable effect optically superposed on at least part of the laser-marking sublayer and a portion devoid of marking with an optically variable effect optically superposed on at least part of the laser-marking sublayer, ∘ transparent to a marking laser radiation, ∘ extending between said outer face and a layer, called a laser-marking sublayer, sensitive to this laser radiation, so as to allow the production of a marking in the marking sublayer under the effect of this laser radiation passing through each covering layer (21, 22), ∘ incorporating at least one marking with an optically variable effect arranged relative to the laser-marking sublayer such that said outer face has a portion with an optically variable effect optically superposed on at least part of the laser-marking sublayer and a portion devoid of marking with an optically variable effect optically superposed on at least part of the laser-marking sublayer, - said at least one covering layer (21, 22) further incorporates at least one marking agent sensitive to said laser radiation so as to allow the production of a marking in said at least one covering layer (21, 22) under the effect of this laser radiation, - said portion devoid of marking with an optically variable effect and said portion with an optically variable effect are produced with contents of at least one marking agent sensitive to said laser radiation, between said outer face of the support and said laser-marking sublayer, which are different, and - said portion devoid of marking with an optically variable effect is devoid of marking agent sensitive to said laser radiation, at least one marking agent sensitive to said laser radiation being incorporated into at least one covering layer (21, 22) only in said portion with an optically variable effect.