The growth, morphology, and magnetic structure of ultrathin Cr films grown on a Ag(001) substrate are studied using low-energy electron diffraction (LEED), angle-resolved photoemission spectroscopy (ARPES), and ab initio density functional theory (DFT) calculations. The presence and temperature dependence of $c(2\ifmmode\times\else\texttimes\fi{}2)$ half-order spots in the LEED pattern, for low electron energies, along with the presence of characteristic Cr $3d$ bands in the ARPES spectra, confirm the existence of antiferromagnetic ordering for the Cr monolayer case. Our experiments are also consistent with the presence of a $p(1\ifmmode\times\else\texttimes\fi{}1)$ Ag overlayer on top of the Cr layer, suggesting the existence of a Ag/Cr/Ag(001) sandwich structure at the surface. Our DFT calculations confirm that this is the most favored geometric and magnetic structure of the system. The Cr layer is found to retain a ``two-dimensional'' character with enhanced Cr $3d$ magnetic moments, despite being buried below a Ag monolayer, due to the absence of significant hybridization between Cr $3d$ and Ag $4d$ electronic states. The coverage dependence of the magnetic ordering indicates a maximum ordering above the expected monolayer coverage, possibly due to intermixing between Ag and Cr atoms in the overlayer.