We report measurements of the hydrostatic pressure dependence of the elastic constants of poly(methyl methacrylate) using high resolution Brillouin spectroscopy. We have measured the density rho(p) up to 500 MPa, and the longitudinal and transverse sound velocities up to 300 MPa. We have calculated the bulk modulus, Youngs modulus, shear modulus and Poisson’s ratio. Initially there is a strong increase in the intensity of the shear phonons which levels off at p > 100 MPa. At this pressure a dispersion-like step in the shear modulus as well as a decrease in the bulk modulus is found, which is tentatively discussed as a ’weak glass transition’ within the glass.