This paper deals with existence of directed balanced incomplete block designs (DBIBDs) in which every block is of size six. The set \(\{(a_ i,a_ j):1 \leq i \leq j \leq k\}\) consisting of \({k \choose 2}\) ordered pairs defines a transitive ordered \(k\)-tuple \((\alpha_ 1, \alpha_ 2, \alpha_ 3,\dots, \alpha_ k)\). A \(\text{DBIBD}(k;\nu; \lambda)\), with positive integers \(\nu,k, \lambda\) as its parameters is a pair \((X,B)\) where \(X\) is a \(\nu\)-set and \(B\) is a collection of transitively ordered \(k\)-tuples of \(X\) (called blocks) such that each ordered pair of points of \(X\) occur in exactly \(\lambda\) blocks of \(B\). It is shown here that the necessary conditions for the existence of a \(\text{DBIBD}(k; \nu; \lambda)\) [these necessary conditions being \(2\lambda (\nu-1) \equiv 0 \pmod{k-1}\) and \(\lambda \nu (\nu-1)\equiv 0\pmod{{k \choose 2}}]\) with \(k=6\) are also sufficient except when \((\nu,\lambda)=(21,1)\).