We present the discovery of a molecular cloud at z abs ≈ 2.5255 along the line of sight to the quasar SDSS J 000015.17+004833.3. We use a high-resolution spectrum obtained with the Ultraviolet and Visual Echelle Spectrograph together with a deep multi-wavelength medium-resolution spectrum obtained with X-shooter (both on the Very Large Telescope) to 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-α system (DLA) with log N(H i) (cm −2 ) = 20.8 ± 0.1. The DLA has supersolar metallicity (Z ∼ 2.5 Z , albeit to within a factor of two to three) with a depletion pattern typical of cold gas and an overall molecular fraction f = 2N(H 2 )/(2N(H 2 ) + N(H i)) ∼ 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 −1 ), cold component in which carbon monoxide molecules are also found, with log N(CO) ≈ 15. With the help of the spectral synthesis code Cloudy, we study the chemical and physical conditions in the cold gas. We find that the line of sight probes the gas deep after the H i-to-H 2 transition in a ∼4−5 pc-size cloud with volumic density n H ∼ 80 cm −3 and temperature of only 50 K. Our model suggests that the presence of small dust grains (down to about 0.001 µm) and high cosmic ray ionisation rate (ζ 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 Å bump. Interestingly, the chemical and physical properties of this cloud are very similar to what is seen in diffuse molecular regions of the nearby Perseus complex, despite the former being observed when the Universe was only 2.5 Gyr old. 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 T CMB (z = 2.53) ≈ 9.6 K, in perfect agreement with the predicted adiabatic cooling of the Universe.