Translation:I give the book to the smartest child.
I think it could be both, genitiv ( of the) or dativ (to the). The context will be important
How do I say: "I give the book of the smartest child?" Move cartea to the end of the sentence?
If you move "cartea" to the end of the sentence ("Eu îi dau celui mai deștept copil cartea."), then we don't really know whose book it is, just that you give that particular book to the smartest child.
The original sentence is ambiguous because you could be giving the book of the smartest child to someone else.