The unique flat connection $\nabla$ with $\nabla ^{0,1}=\overline \partial $ when betti number $b_1(X)=0$.

Question

I came across this question in Huybrechts' Complex Geometry P192 Exercise 4.3.9 and stuck.

Let $X$ be a compact Kähler manifold with $b_1(X)=0$.Show that there exists a unique flat connection $\nabla$ on the trivial holomorphic line bundle $\mathcal O$ with $\nabla ^{0,1}=\overline \partial$.

Note that for a connection $\nabla$ on a vector bundle $E$ is flat ($F_\nabla =0$) if and only if $E$ locally admits trivializing parallel sections.

Any references and suggestion are appreciated.Thanks in advance.