Several high-frequency peaked BL Lac objects such as Mrk 501 are strong TeV emitters. However, a significant fraction of the TeV gamma rays emitted are likely to be absorbed in interactions with the diffuse IR background, yielding electron-positron pairs. Hence, the observed TeV spectrum must be steeper than the intrinsic one. Using the recently derived intrinsic $gamma$-ray spectrum of Mrk 501 during its 1997 high state, we study the inverse-Compton scattering of cosmic microwave photons by the resulting electron-positron pairs, which implies the existence of a hitherto undiscovered GeV emission. The typical duration of the GeV emission is determined by the flaring activity time and the energy-dependent magnetic deflection time. We numerically calculate the scattered photon spectrum for different intergalactic magnetic field (IGMF) strengths, and find a spectral turnover and flare duration at GeV energies which are dependent on the field strength. We also estimate the scattered photon flux in the quiescent state of Mrk 501. The GeV flux levels predicted are consistent with existing EGRET upper limits, and should be detectable above the synchrotron -- self Compton (SSC) component with the {em Gamma-Ray Large Area Space Telescope} ({em GLAST}) for IGMFs $lesssim 10^{-16}$ G, as expected in voids. Such detections would provide constraints on the strength of weak IGMFs.
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