  This paper considers real forms of closed algebraic $\mathbb{C}^*$-embeddings
in $\mathbb{C}^2$. The classification of such embeddings was recently completed
by Cassou-Nogues, Koras, Palka and Russell. Based on their classification, this
paper shows that, up to an algebraic change of coordinates, there is only one
polynomial embedding of the real 1-sphere $\mathbb{S}^1$ in the affine plane
$\mathbb{R}^2$.
