  We study the possible existence of charged and neutral black holes in the
Bergmann-Wagoner class of scalar-tensor theories (STT) of gravity in four
dimensions. The existence of black holes is shown for anomalous versions of
these theories, with a negative kinetic term in the Lagrangian. The Hawking
temperature of these holes is zero, while the horizon area is (in most cases)
infinite. As a special case, the Brans-Dicke theory is studied in more detail,
and two kinds of infinite-area black holes are revealed, with finite and
infinite proper time needed for an infalling particle to reach the horizon;
among them, analyticity properties select a discrete subfamily of solutions,
parametrized by two integers, which admit an extension beyond the horizon. The
causal structure and stability of these solutions with respect to small radial
perturbations is discussed. As a by-product, the stability properties of all
spherically symmetric electrovacuum STT solutions are outlined.
