  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.
