Is a compact kahler manifold with non-zero Neron-Severi groups projective?

Question

In Huybrechts' book 《complex geometry》 p251, there is a statement: A compact Kahler manifold $X$ is projective if and only if $K_X\cap H^2(X,\Bbb Z)\neq 0$, where $K_X$ means the Kahler cone of $X$. And we know Kahler cone is contained in $H^{1,1}(X)$, so we get $K_X\cap H^2(X,\Bbb Z)=K_X\cap H^{1,1}(X)\cap H^2(X,\Bbb Z)=K_X\cap H^{1,1}(X,\Bbb Z)$, we know that $H^{1,1}(X,\Bbb Z)$ is the Neron-Severi group of $X$, so if the Neron-Severi group of $X$ is $0$, we say $X$ can never be projective by the theorem stated above, but on the other hand, if the Neron-Severi group of $X$ is not $0$, can we get the concusion that $X$ must be projective? Or any counter examples?