The edge mode in the fractional quantum Hall effect at ν=2/3 has been the subject of active theoretical and experimental research for decades; yet, its exact structure remains unsettled. We studied the ν=2/3 edge mode structure within a small quantum dot (QD). The conductance plateau of 1/3 e²/h, which appears at transmission half of a single quantum point contact, continue to exist also in the dot geometry with the same total transmission. However, when scanning the Fermi energy inside the QD we observed a series of dips in the conductance, going from the 1/3 e²/h plateau value to almost zero conductance. Our results suggest an edge mode consisting of two edge channels, each with conductance of 1/3 e²/h and tunneling taking place between them. In this case a confined circulating channel is formed by the inner channel, while the outer one passes feely through the QD and acts as in and out leads. At longer propagation lengths (larger QDs), the mixing between the edge channels becomes significant, and a single edge channel with conductance 2/3 e²/h is formed, as observed in macroscopic samples.