  Let $X$ be a compact K{\"a}hler manifold of dimension three. We prove that
there exists a projective manifold $Y$ such that $\pi\_1(X)\simeq \pi\_1(Y)$.
We also prove the bimeromorphic existence of algebraic approximations for
compact K{\"a}hler manifolds of algebraic dimension $\dim(X)-1$. Together with
the work of Graf and the third author, this settles in particular the
bimeromorphic Kodaira problem for compact K{\"a}hler threefolds.
