A significant fraction of the molecular gas in star-forming regions is irradiated by stellar UV photons. In these environments, the electron density (n_e) plays a critical role in the gas dynamics, chemistry, and collisional excitation of certain molecules. We determine n_e in the prototypical strongly irradiated photodissociation region (PDR), the Orion Bar, from the detection of new millimeter-wave carbon recombination lines (mmCRLs) and existing far-IR [13CII] hyperfine line observations. We detect 12 mmCRLs (including alpha, beta, and gamma transitions) observed with the IRAM 30m telescope, at ~25 angular resolution, toward the H/H2 dissociation front (DF) of the Bar. We also present a mmCRL emission cut across the PDR. These lines trace the C+/C/CO gas transition layer. As the much lower frequency carbon radio recombination lines, mmCRLs arise from neutral PDR gas and not from ionized gas in the adjacent HII region. This is readily seen from their narrow line profiles (dv=2.6+/-0.4 km/s) and line peak LSR velocities (v_LSR=+10.7+/-0.2 km/s). Optically thin [13CII] hyperfine lines and molecular lines - emitted close to the DF by trace species such as reactive ions CO+ and HOC+ - show the same line profiles. We use non-LTE excitation models of [13CII] and mmCRLs and derive n_e = 60-100 cm^-3 and T_e = 500-600 K toward the DF. The inferred electron densities are high, up to an order of magnitude higher than previously thought. They provide a lower limit to the gas thermal pressure at the PDR edge without using molecular tracers. We obtain P_th > (2-4)x10^8 cm^-3 K assuming that the electron abundance is equal or lower than the gas-phase elemental abundance of carbon. Such elevated thermal pressures leave little room for magnetic pressure support and agree with a scenario in which the PDR photoevaporates.
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