The electron density ($n_{e^{-}}$) plays an important role in setting the chemistry and physics of the interstellar medium. However, measurements of $n_{e^{-}}$ in neutral clouds have been directly obtained only toward a few lines of sight or they rely on indirect determinations. We use carbon radio recombination lines and the far-infrared lines of C$^{+}$ to directly measure $n_{e^{-}}$ and the gas temperature in the envelope of the integral shaped filament (ISF) in the Orion A molecular cloud. We observed the C$102alpha$ and C$109alpha$ carbon radio recombination lines (CRRLs) using the Effelsberg 100m telescope at ~2 resolution toward five positions in OMC-2 and OMC-3. Since the CRRLs have similar line properties, we averaged them to increase the signal-to-noise ratio of the spectra. We compared the intensities of the averaged CRRLs, and the 158 {mu}m-[CII] and [$^{13}$CII] lines to the predictions of a homogeneous model for the C$^{+}$/C interface in the envelope of a molecular cloud and from this comparison we determined the electron density, temperature and C$^{+}$ column density of the gas. We detect the CRRLs toward four positions, where their velocity and widths (FWHM 2.3 km s$^{-1}$) confirms that they trace the envelope of the ISF. Toward two positions we detect the CRRLs, and the [CII] and [$^{13}$CII] lines with a signal-to-noise ratio >5, and we find $n_{e^{-}}=0.65pm0.12$ cm$^{-3}$ and $0.95pm0.02$ cm$^{-3}$, which corresponds to a gas density $n_{H}approx5times10^{3}$ cm$^{-3}$ and a thermal pressure of $p_{th}approx4times10^{5}$ K cm$^{-3}$. We also constrained the ionization fraction in the denser portions of the molecular cloud using the HCN(1-0) and C$_{2}$H(1-0) lines to $x(e^{-})<3times10^{-6}$. The derived electron densities and ionization fraction imply that $x(e^{-})$ drops by a factor >100 between the C$^{+}$ layer and the regions probed by HCN(1-0).
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