  Molecular beam epitaxy of K-doped Ba122
(Ba$_{1-x}$K$_x$Fe$_\text{2}$As$_\text{2}$) superconductor was realized on a
MgO substrate. Microstructural observation revealed that the undoped Ba122
served as a perfect buffer layer for epitaxial growth of the K-doped Ba122. The
film exhibited a high critical temperature of 39.8 K and a high critical
current density of 3.9 MA/cm$^\text{2}$ at 4 K. The successful growth of
epitaxial thin film will enable artificial single grain boundary on oxide
bicrystal substrates and reveal the grain boundary transport nature of K-doped
Ba122.
