The evolution of a system of cosmic strings is studied using an extended version of an analytic formalism introduced by Kibble. It is shown that, in a radiation-dominated universe, the fate of the string system depends sensitively on the fate of the closed loops that are produced by the interactions of very long strings. The strings can be prevented from dominating the energy density of the Universe only if there is a large probability (> or approx. =50%) that a closed loop will intersect itself and break up into smaller loops. A comparison with the numerical simulations of Albrecht and Turok indicates that the probability of self-intersection is indeed large enough to allow the energy density in strings to stabilize at a small fraction of the radiation density, but there is a potential problem with the gravitational radiation that is produced by the strings. If the string tension is too large, then the gravitational radiation will be so copious that it interferes with primordial nucleosynthesis. By assuming that the probability of self-intersection is less than 85%, as the comparison with the results of Albrecht and Turok indicates, an upper bound on the string tension is obtained: G or approx. =2 x 10 W) predicted for the cosmic-string theory of galaxy formation. This bound would become significantly lower if the probability of intersection is less than 85%.« less