  The string equations of motion and constraints are solved for a ring shaped
Ansatz in cosmological and black hole spacetimes. In FRW universes with
arbitrary power behavior [$R(X^0) = a\;|X^0|^{\a}\, $], the asymptotic form of
the solution is found for both $X^0 \to 0$ and $X^0 \to \infty$ and we plot the
numerical solution for all times. Right after the big bang ($X^0 = 0$), the
string energy decreasess as $ R(X^0)^{-1} $ and the string size grows as $
R(X^0) $ for $ 0 < \a < 1 $ and as $ X^0 $ for $ \a < 0 $ and $ \a > 1 $. Very
soon [ $ X^0 \sim 1 $] , the ring reaches its oscillatory regime with frequency
equal to the winding and constant size and energy. This picture holds for all
values of $ \a $ including string vacua (for which, asymptotically, $ \a = 1$).
In addition, an exact non-oscillatory ring solution is found. For black hole
spacetimes (Schwarzschild, Reissner-Nordstr\oo m and stringy), we solve for
ring strings moving towards the center. Depending on their initial conditions
(essentially the oscillation phase), they are are absorbed or not by
Schwarzschild black holes. The phenomenon of particle transmutation is
explicitly observed (for rings not swallowed by the hole). An effective horizon
is noticed for the rings. Exact and explicit ring solutions inside the
horizon(s) are found. They may be interpreted as strings propagating between
the different universes described by the full black hole manifold.
