  We consider a model for the Higgs sector with two scalar doublets and a
broken $Z_2$ symmetry, the Stealth Doublet Model, where the $Z_2$ symmetry is
manifest in the Yukawa sector but broken by the scalar potential. This model
can be seen as a generalization of the Inert Doublet Model. One of the doublets
is the Higgs doublet that participates in electroweak symmetry breaking and
couples to fermions. The other doublet does not couple to fermions at tree
level and does not acquire a vacuum expectation value. The broken $Z_2$
symmetry leads to interesting phenomenology such as mixing between the two
doublets and charged and CP-odd scalars that can be light and have unusual
decay channels. We present theoretical and experimental constraints on the
model and consider the recent observation of a Higgs boson at the LHC. The data
on the $H\to\gamma\gamma$ channel can be naturally accommodated in the model,
with either the lightest or the heaviest CP-even scalar playing the role of the
observed particle.
