  We propose a reduction procedure that leads to a reduced star product on the
reduced phase space of a `First Class'-constrained system, where no symmetries,
group actions or the like are present. For the case that the coisotropic
submanifold has codimension 1, we establish a constructive method to compute
the reduced star product explicitly. Concluding examples show that this method
depends crucially on the constraint function singled out to describe the
constrained submanifold and not only on this submanifold itself, and that two
different constraint functions for the same constraint submanifold will
generally result in not only different but inequivalent reduced star products.
