We show that for a smooth hypersurface X ⊂ P of degree at least 2, there exist arithmetically Cohen-Macaulay (ACM) codimension two subvarieties Y ⊂ X which are not an intersection X ∩ S for a codimension two subvariety S ⊂ P. We also show there exist Y ⊂ X as above for which the normal bundle sequence for the inclusion Y ⊂ X ⊂ P does not split. Dedicated to Spencer Bloch