  We identify a class of scalar-tensor theories with coupling between the
scalar and the Gauss-Bonnet invariant that exhibit spontaneous scalarization
for both black holes and compact stars. In particular, these theories formally
admit all of the stationary solutions of general relativity, but these are not
dynamically preferred if certain conditions are satisfied. Remarkably, black
holes exhibit scalarization if their mass lies within one of many narrow bands.
We find evidence that scalarization can occur in neutron stars as well.
