We present the discovery of a molecular cloud at zabs=2.5255 along the line of sight to the quasar J0000+0048. We perform a detailed analysis of the absorption lines from ionic, neutral atomic and molecular species in different excitation levels, as well as the broad-band dust extinction. We find that the absorber classifies as a Damped Lyman-alpha system (DLA) with logN(HI)(cm^-2)=20.8+/-0.1. The DLA has super-Solar metallicity with a depletion pattern typical of cold gas and an overall molecular fraction ~50%. This is the highest f-value observed to date in a high-z intervening system. Most of the molecular hydrogen arises from a clearly identified narrow (b~0.7 km/s), cold component in which CO molecules are also found, with logN(CO)~15. We study the chemical and physical conditions in the cold gas. We find that the line of sight probes the gas deep after the HI-to-H2 transition in a ~4-5 pc-size cloud with volumic density nH~80 cm^-3 and temperature of only 50 K. Our model suggests that the presence of small dust grains (down to about 0.001 {mu}m) and high cosmic ray ionisation rate (zeta_H a few times 10^-15 s^-1) are needed to explain the observed atomic and molecular abundances. The presence of small grains is also in agreement with the observed steep extinction curve that also features a 2175 A bump. The properties of this cloud are very similar to what is seen in diffuse molecular regions of the nearby Perseus complex. The high excitation temperature of CO rotational levels towards J0000+0048 betrays however the higher temperature of the cosmic microwave background. Using the derived physical conditions, we correct for a small contribution (0.3 K) of collisional excitation and obtain TCMB(z = 2.53)~9.6 K, in perfect agreement with the predicted adiabatic cooling of the Universe. [abridged]
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