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 nondegeneracy and thereby an imbalance in the population of magnon modes with opposite spins. A temperature gradient then gives rise to a nonzero magnon spin current. This SSE is measured in both a local and a nonlocal geometry at 5 K in bulk NiO. The magnetic field dependence of the obtained signal is modeled 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 nonlocal 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.