  In NiTe$_3$O$_6$ with a chiral crystal structure, we report on a giant
natural optical rotation of the lowest-energy magnon. This polarization
rotation, as large as 140 deg/mm, corresponds to a path difference between
right and left circular polarizations that is comparable to the sample
thickness. Natural optical rotation, being a measure of structural chirality,
is highly unusual for long-wavelength magnons. The collinear antiferromagnetic
order of NiTe$_3$O$_6$ makes this giant effect even more peculiar: Chirality of
the crystal structure does not affect the magnetic ground state but is strongly
manifested in the lowest excited state. We show that the dynamic
magnetoelectric effect, turning this magnon to a magnetic- and electric-dipole
active hybrid mode, generates the giant natural optical rotation. In finite
magnetic fields, it also leads to a strong optical magnetochiral effect.
