  The exact general solution of circular strings in $2+1$ dimensional de Sitter
spacetime is described completely in terms of elliptic functions. The novel
feature here is that one single world-sheet generically describes {\it
infinitely many} (different and independent) strings. This has no analogue in
flat spacetime. The circular strings are either oscillating ("stable") or
indefinitely expanding ("unstable"). We then compute the {\it exact} equation
of state of circular strings in the $2+1$ dimensional de Sitter (dS) and anti
de Sitter (AdS) spacetimes, and analyze its properties for the different
(oscillating, contracting and expanding) strings. We finally perform a
semi-classical quantization of the oscillating circular strings. We find the
mass formula $\alpha'm^2_{\mbox{dS}}\approx 4n-5H^2\alpha'n^2,\;(n\in N_0),$
and a {\it finite} number of states $N_{\mbox{dS}}\approx 0.34/(H^2\alpha')$ in
de Sitter spacetime; $m^2_{\mbox{AdS}}\approx H^2n^2$ (large $n\in N$) and
$N_{\mbox{AdS}}=\infty$ in anti de Sitter spacetime. The level spacing grows
with $n$ in AdS spacetime, while is approximately constant (although smaller
than in Minkowski spacetime and slightly decreasing) in dS spacetime.
