Can any one give an example of a 3-fold X which contains an embedded ample divisor $D \cong CP^2$
with normal bundle $O(3)$ in X?



Looking for a 3-fold with some property?
Let me give an attempt of a proof of the fact that such example does not exist. 


Proof. Suppose $X$ is such a $3$-fold and let $L$ be the line bundle corresponding to the divisor $\mathbb CP^2$. First we will prove that $Pic(X)=\mathbb Z$ and then will get a contradiction. 


Notice that $L$ has a lot of sections. In particular in a neighbourhood of $\mathbb CP^2$ for every two points $x,y$ there is section of $L$ that contains $x$ but does not contain $y$.
And also notice that for every point of $X$ there is a section that does not contain it. Hence we have a morphism $X\to P(H^0(L)^\*)$. This morphism can not contract anything since $L$ is ample. The map is an embedding on the neighbourhood of $D$ and so the image can have singularities at most in codimension $3$.


From this it should follow (I guess), that we can apply Lefshetz that says $Pic (X)=Pic(D)=\mathbb Z$. So $X$ is a Fano with $Pic=\mathbb Z$. Torsten Ekedahl explained how now one can deduce contrudiction, see his comment.


