A half-metal has been defined as a material with propagating electron states at the Fermi energy only for one of the spin directions. But is it fully half-metallic, that is without electrons with opposite spin at that energy? We have studied the spin-conserving process of tunnelling between La0.7Sr0.3MnO3 half-metallic electrodes across an ultrathin SrTiO3 insulator. This experiment demonstrates that the class of half-metallic materials indeed exists at non-zero temperatures, even at interfaces. It also shows that a fully spin-polarized tunnelling current may persist at large bias.