  We study the mechanism of the exceptionally large anomalous Hall effect (AHE)
in the noncentrosymmetric antiferromagnet
$\mathrm{Co}\mathrm{Nb}_3\mathrm{S}_6$ by angle-resolved photoemission
spectroscopy (ARPES) and magnetotransport measurements. From ARPES measurements
of $\mathrm{Co}\mathrm{Nb}_3\mathrm{S}_6$ and its family compounds
($\mathrm{Fe}\mathrm{Nb}_3\mathrm{S}_6$ and
$\mathrm{Ni}\mathrm{Nb}_3\mathrm{S}_6$), we find a band dispersion unique to
the Co intercalation existing near the Fermi level. We further demonstrate that
a slight deficiency of sulfur in $\mathrm{Co}\mathrm{Nb}_3\mathrm{S}_6$
eliminates the ferromagnetism and the AHE simultaneously while hardly changing
the band structure, indicating that the weak ferromagnetism is responsible for
the emergence of the large AHE. Based on our results, we propose Weyl points
near the Fermi level to cause the large AHE.
