  We 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.
The string equation of state has the perfect fluid form $P=(\gamma-1)E,$ with
the pressure and energy expressed closely and completely in terms of elliptic
functions, the instantaneous coefficient $\gamma$ depending on the elliptic
modulus. We semi-classically quantize the oscillating circular strings. The
string mass is $m=\sqrt{C}/(\pi H\alpha'),\;C$ being the Casimir operator,
$C=-L_{\mu\nu}L^{\mu\nu},$ of the $O(3,1)$-dS [$O(2,2)$-AdS] group, and $H$ is
the Hubble constant. We find $\alpha'm^2_{\mbox{dS}}\approx 5.9n,\;(n\in N_0),$
and a {\it finite} number of states $N_{\mbox{dS}}\approx 0.17/(H^2\alpha')$ in
de Sitter spacetime; $m^2_{\mbox{AdS}}\approx 4H^2n^2$ (large $n\in N_0$) and
$N_{\mbox{AdS}}=\infty$ in anti de Sitter spacetime. The level spacing grows
with $n$ in AdS spacetime, while is approximately constant (although larger
than in Minkowski spacetime) in dS spacetime. The massive states in dS
spacetime decay through tunnel effect and the semi-classical decay probability
is computed. The semi-classical quantization of {\it exact} (circular) strings
and the canonical quantization of generic string perturbations around the
string center of mass strongly agree.
