Context : 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 .

Aims : 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 [ ^ { 13 } C ii ] hyperfine line observations .

Methods : We detect 12 mmCRLs ( including \alpha , \beta , and \gamma transitions ) observed with the IRAM 30 m telescope , at \sim 25 ^ { \prime \prime } angular resolution , toward the H / H _ { 2 } dissociation front ( DF ) of the Bar .
We also present a mmCRL emission cut across the PDR .

Results : 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 H ii region .
This is readily seen from their narrow line profiles ( \Delta v = 2.6 \pm 0.4 km s ^ { -1 } ) and line peak velocities ( v _ { LSR } = +10.7 \pm 0.2 km s ^ { -1 } ) .
Optically thin [ ^ { 13 } C ii ] 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 [ ^ { 13 } C ii ] and mmCRLs and derive n _ { e } = 60 – 100 cm ^ { -3 } and T _ { e } = 500 – 600 K toward the DF .

Conclusions : 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 } \geq ( 2 - 4 ) \cdot 10 ^ { 8 } cm ^ { -3 } K assuming that the electron abundance is equal to 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 .