The existence of a (q, k,1) difference family in GF(q) has been completely solved for k=3, 4, 5. For k=6 fundamental results have been given by Wilson. In this article, we continue the investigation and show that the necessary condition for the existence of a(q,6,1) difference family in GF(q), i.e. q ≡ 1 (mod 30) is also sufficient with one exception of q=61. The method of this paper is to lower Wilson's bound by using Weil's theorem on character sums to exploit Wilson's sufficient conditions for the existence of (q,6,1) difference families. The remaining gap is closed by computer searches.