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We have studied the limits of stability in the first order liquid-solid phase transition in a Lennard-Jones system by means of the short-time relaxation method and using the bond-orientational order parameter Q6. These limits are compared with the melting line. We have paid special attention to the supercooled liquid, comparing our results with the point where the free energy cost of forming a nucleating droplet goes to zero. We also indirectly estimate the dimension associated to the critical nucleus at the spinodal, expected to be fractal according to mean field theories of nucleation.