  We classify the possible local holonomy groups of Weyl connections. The
Berger-Simons theorem and the Merkulov-Schwachh\"ofer classification of
holonomy groups of irreducible torsion-free connections leaves us with the
remaining case, where the Weyl connection $D$ is reducible and non-closed. In
this case, it was shown by F. Belgun and A. Moroianu that the Weyl structure is
an adapted Weyl structure of a non-closed conformal product. Furthermore we
prove that non-closed Einstein-Weyl product structures only exist in dimension
$4$.
