Measurements of the equilibrium magnetization curve of type II superconductors have usually only been possible on reversible specimens. In this paper, a method to measure the equilibrium magnetization curve of irreversible type II superconductors is described. Results obtained by this method from a hysteretic Nb specimen exhibiting the peak effect are reported. They show an irregularity in the shape of the equilibrium magnetization curve in the neighborhood ofH c2 which corresponds to a minimum of the compressional modulusK of the flux line lattice. The minimum ofK also corresponds with the maximum of the critical current densityj c, which strongly suggests that in this case the peak effect is originated by a “soft point” of the flux line lattice. The observed anomaly ofK is discussed in terms of a possible first-order phase transition in the flux line lattice.