  We report the observation of magnon spin currents generated by the Spin
Seebeck effect (SSE) in a bulk single crystal of the easy-plane antiferromagnet
NiO. A magnetic field induces a non-degeneracy and thereby an imbalance in the
population of magnon modes with opposite spin. A temperature gradient then
gives rise to a non-zero magnon spin current. This SSE is measured both in a
local and a non-local geometry at 5$\,$K in bulk NiO. The magnetic field
dependence of the obtained signal is modelled by magnetic field splitting of
the low energy magnon modes, affecting the spin Seebeck coefficient. The
relevant magnon modes at this temperature are linked to cubic anisotropy and
magnetic dipole-dipole interactions. The non-local signal deviates from the
expected quadratic Joule heating by saturating at a current from around
75$\,\mu A$ in the injector. The magnon chemical potential does not decay
exponentially with distance and inhomogeneities may be the result of local
magnon accumulations.
