  The hexagonal swedenborgite, CaBaCo$_2$Fe$_2$O$_7$, is a chiral frustrated
antiferromagnet, in which magnetic ions form alternating Kagome and triangular
layers. We observe a long range $\sqrt{3} \times \sqrt{3}$ antiferromagnetic
order setting in below $T_N = 160$ K by neutron diffraction on single crystals
of CaBaCo$_2$Fe$_2$O$_7$. Both magnetization and polarized neutron single
crystal diffraction measurements show that close to $T_N$ spins lie
predominantly in the $ab$-plane, while upon cooling the spin structure becomes
increasingly canted due to Dzyaloshinskii-Moriya interactions. The ordered
structure can be described and refined within the magnetic space group
$P31m^\prime$. Diffuse scattering between the magnetic peaks reveals that the
spin order is partial. Monte Carlo simulations based on a Heisenberg model with
two nearest-neighbor exchange interactions show a similar diffuse scattering
and coexistence of the $\sqrt{3} \times \sqrt{3}$ order with disorder. The
coexistence can be explained by the freedom to vary spins without affecting the
long range order, which gives rise to ground-state degeneracy. Polarization
analysis of the magnetic peaks indicates the presence of long-period cycloidal
spin correlations resulting from the broken inversion symmetry of the lattice,
in agreement with our symmetry analysis.
