It is currently possible to compute reliably the free energy of a material at finite temperature and pressure from the quantum mechanical ground state of the electrons within density functional theory. The free energy is obtained by thermodynamic integration in the context of ab-initio molecular dynamics. By applying this approach to coexisting liquid and solid phases, one can predict a melting line from first-principles microscopic theory. In this talk I will review the basic computational approach, and will discuss how different approximations for the exchange and correlation functional affect the predicted phase diagram. Finally I will present a recent application in which this scheme has been used to determine the diamond melting line in an extended range of temperatures and pressures. This coexistence line is poorly known from experiment, given the extreme pressures and temperatures that are required to melt diamond. The implications of the newly predicted carbon phase diagram will be discussed.