A balanced ternary design with V elements is a collection of B blocks (which are multisets) of size K such that each element occurs 0, 1 or 2 times per block and R times altogether, and such that each unordered pair of distinct elements occur \(\Lambda\) times. Then each element has to occur singly in a constant number \(\rho_ 1\) o blocks and so each element also occurs twice in a constant number \(\rho_ 2\) of blocks, where \(R=\rho_ 1+2\rho_ 2\). If \(\rho_ 2=0\), the design is a balanced incomplete block design (binary design). For non-degeneracy, it is assumed that \(\rho_ 2>0\) and \(K1\) if \(\rho_ 2>0\) (and \(K>2).\) In this paper, the author gives necessary and sufficient conditions for the existence of a balanced ternary design with \(K=3\), and \(\Lambda >1\) and any \(\rho_ 2>0\).