  Here we demonstrate the controlled growth of Bi(110) and Bi(111) films on an
(insulating) $\alpha$-Al$_2$O$_3$(0001) substrate by surface X-ray diffraction
and X-ray reflectivity using synchrotron radiation. At temperatures as low as
40 K, unanticipated pseudo-cubic Bi(110) films are grown having a thickness
ranging from a few to tens of nanometers. The roughness at the film-vacuum as
well as at the film-substrate interface, can be reduced by mild heating, where
a crystallographic orientation transition of Bi(110) towards Bi(111) is
observed at 400 K. From 450 K onwards high quality and ultrasmooth Bi(111)
films are formed. Growth around the transition temperature results in the
growth of competing Bi(110) and Bi(111) thin film domains.
