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We explicitly determine the elliptic K3 surfaces with a maximal singularfibre. If the characteristic of the ground field is different from 2, for eachof the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surfaceis unique. In characteristic 2 the maximal fibre types are $I_{18}$ and$I^*_{13}$, and there exist two (resp. one) one-parameter families of suchsurfaces.

On the uniqueness of elliptic K3 surfaces with maximal singular fibre