We use a hydrodynamic simulation of an inflationary cold dark matter model with a cosmological constant to predict properties of the extragalactic X-ray background (XRB). We focus on emission from the intergalactic medium (IGM), with particular attention to diffuse emission from warm-hot gas that lies in relatively smooth filamentary structures between galaxies and galaxy clusters. We also include X-rays from point sources associated with galaxies in the simulation, and we make maps of the angular distribution of the emission. Although much of the X-ray luminous gas has a filamentary structure, the filaments are not evident in the simulated maps because of projection effects. In the soft (0.5-2 keV) band, our calculated mean intensity of radiation from intergalactic and cluster gas is 2.3 × 10-12 ergs-1 cm-2 deg-2, 35% of the total softband emission. This intensity is compatible at the ~1 σ level with estimates of the unresolved soft background intensity from deep ROSAT and Chandra measurements. Only 4% of the hard (2-10 keV) emission is associated with intergalactic gas. Relative to active galactic nuclei flux, the IGM component of the XRB peaks at a lower redshift (median z ~ 0.45) and spans a narrower redshift range, so its clustering makes an important contribution to the angular correlation function of the total emission. The clustering on the scales accessible to our simulation (01-10') is significant, with an amplitude roughly consistent with an extrapolation of recent ROSAT results to small scales. A cross-correlation analysis of the XRB against nearby galaxies taken from a simulated redshift survey also yields a strong signal from the IGM. Our conclusions about the soft background intensity differ from those of some recent papers that have argued that the expected emission from gas in galaxy, group, and cluster halos would exceed the observed background unless much of the gas is expelled by supernova feedback. We obtain reasonable compatibility with current observations in a simulation that incorporates cooling, star formation, and only modest feedback. A clear prediction of our model is that the unresolved portion of the soft XRB will remain mostly unresolved even as observations reach deeper point-source sensitivity.