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 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 phenomenon 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 (S > 0): E not equal 0 even at S = 0, E decreases for small S and increases a S for large S. For an oscillating string (0 less than or equal to S less than or equal to 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. For each b, the two independent solutions S-+ and S-- are analyzed. For b < 1/4, all the strings of the S-- solution are unstable (S-max = infinity) and never collapse to a point (S-min not equal 0). S-+ describes one stable (S-max is bounded) oscillating string and (E) is an increasing function of b for 0 less than or equal to b less than or equal to 1/4. For b > 1/4, all strings (for both S-+ and S--) are unstable and have a collapse during their evolution. For b = 1/4, S-- describes two strings (one stable and one unstable for large de Sitter radius), while S-+ describes one stable non-oscillating string.
