By means of Monte Carlo computer simulation and scaling theory, we study the domain growth kinetics associated with a weak first-order transition between two non-symmetry-related ordered phases, exemplified by martensitic transformations, surface reconstructions, or magnetic transitions. The model studied has two kinds of domain walls: sharp, straight stacking faults, and broad, curved solitonlike walls. The domain wall motion after a deep quench to low temperature is found to follow the Allen-Cahn theory; nonetheless, the growth exponent for the excess energy ΔE(t)∼ t − n has an exponent n∼(1/4, distinctly lower than the expected n=(1/2, but in agreement with simulation results for some other models. A theoretical scaling analysis gives exactly n=(1/4 and shows that the slow growth is a consequence of the fact that finite-size stacking faults cannot move until their extent is sufficiently small. The new universality class for domain growth is proposed as a singular Allen-Cahn class with n=(1/4 for nonconserved order parameter, with domain walls of both exactly zero and finite curvature (whereas the domain-wall width or softness is not important as such). Since stacking faults and twin boundaries are common and have exactly zero curvature, we expect that many experimental systems belong to this class. The simulation results are also analyzed in terms of a soliton model as well as the Ginzburg-Landau theory; finally, a fast algorithm for domain growth studies is described.