The exchange bias of the magnetic hysteresis loop and the exchange-spring effect have been observed in thin-film heterostructures comprising layers of Pd_0.96Fe_0.04 low-temperature ferromagnet, metallic cobalt, and cobalt monoxide (CoO) grown on single-crystalline sapphire by methods of magnetron sputtering and molecular beam epitaxy in an ultrahigh-vacuum system. It is established that the exchange anisotropy of Pd_0.96Fe_0.04 layer in Pd_0.96Fe_0.04/CoO/Co/Al_2O_3 thin-film heterostructure is absent and the exchange-spring effect in this system is not manifested. In the case of a Pd_0.96Fe_0.04/Co/CoO/Al_2O_3 structure, it is shown that, according to the results of magnetometric measurements, the inversion of the order of antiferromagnetic and ferromagnetic layers leads to pinning of the magnetic moment of Pd_0.96Fe_0.04 layer at the interface with cobalt and to realization of the exchange-spring effect. Additional evidence in favor of this interpretation was obtained using the micromagnetic modeling of magnetic hysteresis loops.