The structural characteristics of an epitaxial $BaTiO_3$ film on MgO(001) grown by sputtering were studied as a function of temperature using in-situ, real time synchrotron x-ray scattering experiments. We found that the as-grown film was single c-domain but strained at room temperature and tetragonally distorted with the c-axis normal to the film surface. Interestingly, its lattice parameters were found to be expanded in both the in-plane and the out-of -plane directions, i.e. biaxially, comparing with those of a bulk $BaTiO_3$ . More importantly, as it was heated up to $600^{circ}C$, the tetragonal structure was kept up through without and any phase transition, which is usually observed in other epitaxial ferroelectric thin films.