Massive stars can determine the evolution of molecular clouds by eroding and photo-evaporating their surfaces with strong ultraviolet (UV) radiation fields. Moreover, UV radiation is relevant in setting the thermal gas pressure in star-forming clouds, whose influence can extend across various spatial scales, from the rims of molecular clouds to entire star-forming galaxies. Probing the fundamental structure of nearby molecular clouds is therefore crucial to understand how massive stars shape their surrounding medium and how fast molecular clouds are destroyed, specifically at their UV-illuminated edges, where models predict an intermediate zone of neutral atomic gas between the molecular cloud and the surrounding ionized gas whose size is directly related to the exposed physical conditions. We present the highest angular resolution (~0.″5, corresponding to 207 au) and velocity-resolved images of the molecular gas emission in the Horsehead nebula, using CO J = 3–2 and HCO⁺ J = 4−3 observations with the Atacama Large Millimeter/submillimeter Array (ALMA). We find that CO and HCO⁺ are present at the edge of the cloud, very close to the ionization (H⁺/H) and dissociation fronts (H/H₂), suggesting a very thin layer of neutral atomic gas (<650 au) and a small amount of CO-dark gas (A_V = 0.006–0.26 mag) for stellar UV illumination conditions typical of molecular clouds in the Milky Way. The new ALMA observations reveal a web of molecular gas filaments with an estimated thermal gas pressure of P_th = (2.3 – 4.0) × 10⁶ K cm⁻³, and the presence of a steep density gradient at the cloud edge that can be well explained by stationary isobaric photo-dissociation region (PDR) models with pressures consistent with our estimations. However, in the H II region and PDR interface, we find P_th,PDR > P_{th,H II} suggesting the gas is slightly compressed. Therefore, dynamical effects cannot be completely ruled out and even higher angular observations will be needed to unveil their role.