We predict the spectrum and flux of the high-energy diffuse cosmic gamma-ray background radiation produced by blazars. Our calculations are based on the Compton Observatory EGRET observations of high-energy gamma-rays from these objects, the Whipple observation of TeV gamma-rays from Mrk 421, and radio population studies of flat-spectrum radio sources. Our statistical analysis gives the result L(gamma) is-proportional-to L(radio)1.2 +/- 0.1, consistent with a linear correlation. We then use the radio luminosity function and redshift evolution to calculate the gamma-ray background. We find that blazars cannot account for the flux level or steep spectrum of the background observed by SAS 2 in the sub-GeV range, but could likely provide the dominant background component in the multi-GeV energy range and above. We predict a differential photon spectral index for this component of approximately 2 up to approximately 10 GeV. Above that energy, the spectrum should gradually steepen owing to pair production by interactions of the gamma-rays with intergalactic infrared photons. Using the calculations of this effect by Stecker, de Jager, & Salamon, we estimate a spectral index of approximately 3.5 for energies above a TeV.