We report a measurement of the neutron lifetime using ultracold neutrons stored in a magneto-gravitational trap made of permanent magnets. Neutrons surviving in the trap after fixed storage times have been counted and the trap losses have continuously been monitored during storage by detecting neutrons leaking from the trap. The value of the neutron lifetime resulting from this measurement is τ_n = (878.3 ± 1.6_stat ± 1.0_syst) A unique feature of this experiment is the monitoring of leaking neutrons providing a robust control of the main systematic loss.