As Sandor points out, this is Stein factorization and the connectedness theorem of Zariski. Let $X$, $Y$ be varieties over a field $K$. Let $h: X\to Y$ be a proper morphism. Then $h_*(O_X)$ is coherent and $Y':=Spec(h_*{\mathcal{O}}_X)$ (cf. EGA II for the definition of $Spec$ of a sheaf of quasicoherent algebras) is finite over $Y$. Consider the Stein factorization $$X\buildrel h' \over\longrightarrow Y'\buildrel g\over \longrightarrow Y.$$ Then (cf. EGA III.4.3.1 - III.4.3.4) $h'$ has non-empty geometrically connected fibres (and $g$ is finite).
Now assume that $h$ is an alteration, i.e. that $h$ is surjective and that there exists a nonempty open subscheme $U\subset Y$ such that $h^{-1}(U)\to U$ is finite. Replacing $U$ by a smaller set we may assume that $U=Spec(A)$ is affine. Define $U':=g^{-1}(U)$.