We have observed a large exchange bias field HE ≈ 2460 Oe and a large coercive field HC ≈ 6200 Oe at T = 2 K for Co/CoO core-shell nanoparticles (~4 nm diameter Co metal core and CoO shell with ~1 nm thickness) embedded in a non-magnetic MgO matrix. Our results are in sharp contrast to the small exchange bias and coercive field in the case of a non-magnetic Al2O3 or C matrix materials reported in previous studies. Using soft X-ray magnetic circular dichroism at the Co-L2,3 edge, we have observed a ferromagnetic signal originating from the antiferromagnetic CoO shell. This gives direct evidence for the existence of rotatable interfacial uncompensated Co spins in the nominally antiferromagnetic CoO shell, thus supporting the uncompensated spin model as a microscopic description of the exchange bias mechanism.