We investigate the motion of an interface between a nematic liquid crystal phase and the isotropic phase of the same fluid. In this simplified model we assume the nematic liquid crystal to have one order parameter only and also suppose the system to be isothermal and initially quenched into the metastable régime of the isotropic phase $(T_{\rm NI}>T>T^*)$. What results is the time-dependent Ginzburg-Landau equation, with domain wall solutions, corresponding to phase interfaces, which interpolate between static isotropic and nematic minima. These domain walls move with a unique velocity which depends more or less linearly on the degree of undercooling. For real liquid crystals this velocity is of the order of cm s$^{-1}$. We also examine the relaxation mode solutions of the Ginzburg-Landau equation, and present a complete phase-diagram of these solutions.