An arc (or independent set) of a Steiner triple system is a set of points containing no triple. An arc is called complete (or maximal) if it is not contained in a larger arc. If a complete arc of an \(S(2,3,v)\) has size \(s\), then one must have \(s(s+ 1)/2\geq v\). Given that a necessary and sufficient condition for the existence of an \(S(2,3,v)\) is \(v\equiv 1\) or \(3\pmod 6\), equality can only occur for \(s\equiv 1\) or \(2\pmod 4\). C. J. Colbourn, K. T. Phelps, M. J. de Resmini and A. Rosa [Discrete Math. 89, No. 2, 149-160 (1991; Zbl 0766.05011)] classified all \(S(2,3,s(s+ 1)/2)\) with \(s\equiv 1\pmod 4\) that can be partitioned into \((s+1)/2\) complete arcs of size \(s\). They asked whether there exist any \(S(2,3,s(s+ 1)/2)\) with \(s\equiv 2\pmod 4\) that contain \(s/2\) disjoint complete arcs of size \(s\). The author considers the smallest possible value of \(s\) (namely 6) and constructs an \(S(2,3,21)\) with 3 disjoint complete arcs of size 6, thereby answering the question in the positive.