  The exact general evolution of circular strings in $2+1$ dimensional de
Sitter spacetime is described closely and completely in terms of elliptic
functions. The evolution depends on a constant parameter $b$, related to the
string energy, and falls into three classes depending on whether $b<1/4$
(oscillatory motion), $b=1/4$ (degenerated, hyperbolic motion) or $b>1/4$
(unbounded motion). The novel feature here is that one single world-sheet
generically describes {\it infinitely many} (different and independent)
strings. The world-sheet time $\tau$ is an infinite-valued function of the
string physical time, each branch yields a different string. This has no
analogue in flat spacetime. We compute the string energy $E$ as a function of
the string proper size $S$, and analyze it for the expanding and oscillating
strings. For expanding strings $(\dot{S}>0)$: $E\neq 0$ even at $S=0$, $E$
decreases for small $S$ and increases $\propto\hspace*{-1mm}S$ for large $S$.
For an oscillating string $(0\leq S\leq S_{max})$, the average energy $<E>$
over one oscillation period is expressed as a function of $S_{max}$ as a
complete elliptic integral of the third kind.
