We investigate SU(3)-periodic vortices in the self-dual Chern-Simons theory
proposed by Dunne in [13, 15]. At the first admissible non-zero energy lev
el E = 2 pi, and for each (broken and unbroken) vacuum state phi((0)) of th
e system, we find a family of periodic vortices asymptotically gauge equiva
lent to phi((0)), as the Chern-Simons coupling parameter k -> 0. At higher
energy levels, we show the existence of multiple gauge distinct periodic vo
rtices with at least one of them asymptotically gauge equivalent to the (br
oken) principal embedding vacuum, when k -> 0.