We present a new method for probing the thermal electron content of the Galaxy by spectral analysis of background point sources in the absorption-only limit to the radiative transfer equation . In this limit , calculating the spectral index , \alpha , of these sources using a natural logarithm results in an additive factor , which we denote \alpha _ { EM } , resulting from the absorption of radiation due to the Galactic thermal electron population . We find that this effect is important at very low frequencies ( \nu \lesssim 200 MHz ) , and that the frequency spacing is critical . We model this effect by calculating the emission measure across the sky . A ( smooth ) thermal electron model for the Galaxy does not fit the observed emission measure distribution , but a simple , cloud-based model to represent the clumpy nature of the warm interstellar medium does . This model statistically reproduces the Galactic emission measure distribution as obtained independently from H \alpha data well . We find that at the lowest frequencies ( \sim 10 - 50 MHz ) , the observed spectral index for a large segment of the Galaxy below Galactic latitudes of \lesssim 15 ^ { \circ } could be changed significantly ( i.e. , \mbox { $ \alpha _ { EM } $ } \gtrsim 0.1 ) . This method therefore provides a correction to low-frequency spectral index measurements of extragalactic sources , and provides a sensitive probe of the thermal electron distribution of the Galaxy using current and next-generation low-frequency radio telescopes . We show that this effect should be robustly detectable individually in the strongest sources , and statistically in source samples at a level of \mbox { $ \alpha _ { EM } $ } \gtrsim 0.18 , 0.06 , and 0.02 for source densities of 10 , 100 and 1,000 sources per square degree .