We analyzed absorption features arising from interstellar neutral carbon that appeared in the UV spectra of 89 stars recorded in the highest resolution echelle modes of the Space Telescope Imaging Spectrograph on board the Hubble Space Telescope so that we could determine the relative populations of collisionally excited fine-structure levels in the atom's electronic ground state. From this information, in combination with molecular hydrogen rotation temperatures, we derive the distribution of thermal pressures in the diffuse, cold neutral medium (CNM). We find a lognormal pressure distribution (weighted by mass) with a mean in log(p/k) equal to 3.58 and an rms dispersion of at least 0.175dex that plausibly arises from turbulence with a characteristic Mach number in the range 1<M<4.