We study the nucleation of the stable phase in a time-dependent Ginzburg-Landau model with an asymmetric double-well structure. The system is taken to be initially in the metastable phase and is then driven toward stable equilibrium by thermal noise or by randomness in the initial conditions. Results are presented as a function of the field h, which splits a pair of degenerate minima, and the noise strength or dispersion in the initial field values. Over the rather wide range of parameter values that we have considered, we find that the nucleation rate can be fairly accurately described by an activated form. The activation energy is a stronger function of h for intermediate values of h than predicted by classical theory, and it is a weak analytic function of noise strength.