The algebraic number fields of degree 6 having Galois group
$S_5$ and minimum discriminant are determined for signatures
(0,\,3), (2,\,2) and (6,\,0). The fields $F_0$, $F_2$, $F_6$
are generated by roots of
$f_0(t) = t^6 + 3 t^4 + 2 t^3 + 6 t^2 + 1$,
$\,f_2(t) = t^6 - 2 t^4 + 12 t^3 - 16 t + 8$, and
$f_6(t) = t^6 - 18 t^4 + 9 t^3 + 90 t^2 - 70 t - 69$
respectively. Each of these fields is unique up to isomorphism.
This completes the enumeration of primitive sextic fields with
minimum discriminant for all possible combinations of Galois group
and signature.