  The second order Ordinary Differential Equation which describes the unknown
part of the solution space of some vacuum Bianchi Cosmologies is completely
integrated for Type III, thus obtaining the general solution to Einstein's
Field Equations for this case, with the aid of the sixth Painlev\'{e}
transcendent $P_{VI}$. For particular representations of $P_{VI}$ we obtain the
known Kinnersley two-parameter space-time and a solution of Euclidean
signature. The imposition of the space-time generalization of a "hidden"
symmetry of the generic Type III spatial slice, enables us to retrieve the
two-parameter subfamily without considering the Painlev\'{e} transcendent.
