  We consider a free-fermion chain with a conformal defect that features an
extended zero mode, and study the entanglement properties in its mixed ground
state. The zero-mode induced degeneracy modifies the density of states in the
single-particle entanglement spectrum, which can be calculated via the full
counting statistics. For a homogeneous chain, the resulting change in the
R\'enyi entropy is derived analytically for arbitrary subsystem ratios in the
thermodynamic limit. For a conformal defect located in the center, analogous
results can be obtained for the half-chain entanglement. In particular, we
observe parity effects for half-chains with even/odd sites, which do not decay
with size.
