The density-functional study of the magnetic properties of a single Fe monolayer on W(001) is reported. The system possesses, presently unique for Fe layers, an antiferromagnetic structure. We demonstrate that the study of the instability of the ferromagnetic state predicts the ground-state antiferromagnetic structure. We use the frozen-magnon technique to evaluate the interatomic exchange parameters of different referent states. The relative role of the Fe-Fe and Fe-W exchange interactions is revealed. The Néel temperature is estimated.