Near-infrared images of the Galactic bulge at 1.25, 2.2, 3.5, and 4.9 mu m obtained by the Diffuse Infrared Background Experiment (DIRBE) onboard the Cosmic Background Explorer (COBE) satellite are used to characterize its morphology and to determine its infrared luminosity and mass. Earlier analysis of the DIRBE observations (Weiland et al. 1994) provided supporting evidence for the claim made by Blitz and Spergel (1991) that the bulge is bar-shaped with its near end in the first Galactic quadrant. Adopting various triaxial analytical functions to represent the volume emissivity of the source, we confirm the barlike nature of the bulge and show that triaxial Gaussian-type functions provide a better fit to the data than other classes of functions, including an axisymmetric spheroid. The introduction of a ''boxy'' geometry, such as the one used by Kent, Dame, and Fazio (1991) improves the fit to the data. Our results show that the bar is rotated in the plane with its near side in the first Galactic quadrant creating an angle of 20 degrees + 10 degrees between its major axis and the line of sight to the Galactic center. Typical axis ratios of the bar are {1:0.33 +/- 0.11:0.23 +/- 0.08}, resembling the geometry of prolate spheroids. There is no statistically significant evidence for an out-of-plane tilt of the bar at 2.2 mu m, and marginal evidence for a tilt of approximate to 2 degrees at 4.9 mu m. The introduction of a roll around the intrinsic major axis of the bulge improves the ''boxy'' appearance of some functions. A simple integration of the observed projected intensity of the bulge gives a bulge luminosity of 1.2 x 10(9), 4.1 x 10(8), 2.3 x 10(8), and 4.3 x 10(7) L., respectively, at 1.25, 2.2, 3.5, and 4.9 mu m wavelength for a Galactocentric distance of 8.5 kpc. The 2.2 mu m luminosity function of the bulge population in the direction of Baade's window yields a bolometric luminosity of L(bol) = 5.3 x 10(9) L.. Stellar evolutionary models relate this luminosity to the number of main-sequence progenitor stars that currently populate the red giant branch. Combined with the recent determination of the main-sequence turnoff mass for the bulge by the Hubble Space Telescope (Hortzman et al. 1993) we derive a photometrically determined bulge mass of approximate to 1.3 x 10(10) M. for a Salpeter initial mass function extended down to 0.1 M..