Recently, the Fermi Space Telescope discovered two large γ-ray emission regions, the so-called Fermi bubbles, that extend up to ~50° above and below the Galactic center (GC). The γ-ray emission from the bubbles is found to follow a hard spectrum with no significant spatial variation in intensity and spectral shape. The origin of the emission is still not clearly understood. Suggested explanations include the injection of cosmic-ray (CR) nuclei from the GC by high-speed Galactic winds, electron acceleration by multiple shocks, and stochastic electron acceleration inside the bubbles. In this Letter, it is proposed that the γ-rays may be the result of diffusive injection of Galactic CR protons during their propagation through the Galaxy. Considering that the bubbles are slowly expanding, and CRs undergo much slower diffusion inside the bubbles than in the average Galaxy and at the same time suffer losses due to adiabatic expansion and inelastic collisions with the bubble plasma, this model can explain the observed intensity profile, the emission spectrum and the measured luminosity without invoking any additional particle production processes, unlike other existing models.