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.
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