Summary: Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by C. Ding and T. Helleseth [IEEE Trans. Inf. Theory 59, No. 9, 5898–5904 (2013; Zbl 1364.94652)]. Let \(\mathcal{C}_{(1, e)}\) be a cyclic code of length \(3^m - 1\) over \(\mathrm{GF}(3)\) with two nonzeros \(\alpha\) and \(\alpha^e\), where \(m\) and \(e\) are given integers, and \(\alpha\) is a generator of \(\mathrm{GF}(3^m)^*\). It is shown that \(\mathcal{C}_{(1, e)}\) is optimal with parameters \([3^m - 1, 3^m - 1 - 2 m, 4]\) if one of the following conditions is met. 1) \(m \equiv 0\pmod 4\), \(m \ge 4\), and \(e = 3^{\frac{m}{2}} + 5. 2) m \equiv 2\pmod 4\), \(m \ge 6\), and \(e = 3^{\frac{m + 2}{2}} + 5\).