We study the response of the one-dimensional charge density wave in K_{0.3}MoO_{3} to different types of excitation with femtosecond optical pulses. We compare direct excitation of the lattice at midinfrared frequencies with injection of quasiparticles across the low energy charge density wave gap and with charge transfer excitation in the near infrared. For all three cases, we observe a fluence threshold above which the amplitude-mode oscillation frequency is softened and the mode becomes increasingly damped. We show that all the data can be collapsed onto a universal curve in which the melting of the charge density wave occurs abruptly at a critical lattice excursion. These data highlight the existence of a universal stability limit for a charge density wave, reminiscent of the Lindemann criterion for the melting of a crystal lattice.