The phase diagram of the organic superconductor κ-(BEDT-TTF)2Cu[N(CN)2Cl has been investigated by ultrasonic velocity measurements under helium gas pressure. Different phase transitions were identified trough several elastic anomalies characterized from isobaric and isothermal sweeps. Our data reveal two crossover lines that end on the critical point terminating the first-order Mott transition line. When the critical point is approached along these lines, we observe a dramatic softening of the velocity which is consistent with a diverging compressibility of the electronic degrees of freedom.PACS numbers: 74.70. Kn, 71.30.+h, 74.25.Ld Electronic properties of metals or semiconductors at energies of the order of room temperature or less can usually be understood by simple models that take into account a few electronic bands near the Fermi level. In the absence of broken symmetries, it suffices to take into account the effects on the bands of small residual electronelectron and electron-phonon interactions to explain the observed electronic properties. In the last decade, a flurry of activity has centered on materials that cannot be understood using the above textbook procedure. In these materials, only one or a few bands should be relevant to understand electronic properties but the residual interactions are so strong that electronic eigenstates can become localized, leading to a complete breakdown of the band picture. A few compounds have become prototype materials for the study of these so-called strong correlation effects:In these compounds, one observes a pressure-induced finite-temperature first-order phase transition (MI) between an insulating (I) and a metallic (M) phase. Pressure (hydrostatic or chemical) increases the bandwidth, reducing the effects of residual interactions. This firstorder phase transition seems to correspond to the socalled Mott transition, as it has become understood in the last few years through dynamical mean-field theory (DMFT) [4].One of the recent predictions of DMFT, is that the compressibility of electronic degrees of freedom diverges at the critical point that terminates the first-order Mott transition line [5,6]. Observation of this phenomenon would help confirm the picture of the Mott transition proposed by DMFT. In this letter, we present the first ultrasonic study of κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl (denoted as κ-Cl), using an hydrostatic helium gas pressure cell. The study, summarized in the phase diagram shown in figure 1, shows (a) A very large softening of the sound velocity at the critical point, corresponding to the predicted divergence of the compressibility of the electronic degrees of freedom. (b) Two crossover lines -joining at the critical point -where a similar although smaller compressibility anomaly is observed. Most remarkably, while the compressibility anomalies decrease in size as one moves away from the critical point, the crossover line at high pressure coincides with the well-known pseudogap features identified in the magnetic [7], transport [8,9] and el...