  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 $\alpha_\mathrm{EM}$, resulting from the
absorption of radiation due to the Galactic thermal electron population. We
find that this effect is important at very low frequencies ($\nu\lesssim200$
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 ($\sim10-50$ MHz), the observed
spectral index for a large segment of the Galaxy below Galactic latitudes of
$\lesssim15^\circ$ could be changed significantly (i.e.,
$\alpha_\mathrm{EM}\gtrsim0.1$).
  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
$\alpha_\mathrm{EM}\gtrsim0.18,0.06$, and 0.02 for source densities of 10, 100
and 1,000 sources per square degree.
