The paraelectric cubic-to-ferroelectric tetragonal phase transition of the thin Pb(Zr, Ti)$O_3$ (PZT) films grown on MgO(001) substrate was investigated in a series of synchrotron x-ray scattering experiments. As the thickness of the film decreases the transition temperature and the amount of the tetragonal distortion were decreased continuously Different from only the c-domains were existent in the thinnest 25nm thick film. Based on this we propose a model for the domain structure of the tetragonal PZT/MgO(100) film that is very different from the ones suggested in literature. We attribute the suppression of the transition to the substrate field that prefers the c-type domains near the interface and suppresses the tetragonal distortion to minimize the film-substrate lattice mismatch.