The structure and properties of a newly emerged solar active region (NOAA Active Region 7985) are discussed using the Coronal Diagnostic Spectrometer (CDS) and the Extreme-Ultraviolet Imaging Telescope (EIT) on board the Solar and Heliospheric Observatory. CDS obtained high-resolution EUV spectra in the 308-381 Å and 513-633 Å wavelength ranges, while EIT recorded full-disk EUV images in the He II (304 Å), Fe IX/X (171 Å), Fe XII (195 Å), and Fe XV (284 Å) bandpasses. Electron density measurements from Si IX, Si X, Fe XII, Fe XIII, and Fe XIV line ratios indicate that the region consists of a central high-density core with peak densities of the order of 1.2×1010 cm-3, which decrease monotonically to ~5.0×108 cm-3 at the active region boundary. The derived electron densities also vary systematically with temperature. Electron pressures as a function of both active region position and temperature were estimated using the derived electron densities and ion formation temperatures, and the constant pressure assumption was found to be an unrealistic simplification. Indeed, the active region is found to have a high-pressure core (1.3×1016 cm-3 K) that falls to 6.0×1014 cm-3 K just outside the region. CDS line ratios from different ionization stages of iron, specifically Fe XVI (335.4 Å) and Fe XIV (334.4 Å), were used to diagnose plasma temperatures within the active region. Using this method, peak temperatures of 2.1×106 K were identified. This is in good agreement with electron temperatures derived using EIT filter ratios and the two-temperature model of Zhang et al. The high-temperature emission is confined to the active region core, while emission from cooler (1-1.6)×106 K lines originates in a system of loops visible in EIT 171 and 195 Å images. Finally, the three-dimensional geometry of the active region is investigated using potential field extrapolations from a Kitt Peak magnetogram. The combination of EUV and magnetic field extrapolations extends the ``core-halo'' picture of active region structure to one in which the core is composed of a number of compact coronal loops that confine the hot, dense, high-pressure core plasma while the halo emission emerges from a system of cooler and more extended loops.