  The existence of a dilaton (or moduli) with gravitational-strength coupling
to matter imposes stringent constraints on the allowed energy scale of cosmic
strings, $\eta$. In particular, superheavy gauge strings with $\eta \sim
10^{16} GeV$ are ruled out unless the dilaton mass $m_{\phi} \gsim 100 TeV$,
while the currently popular value $m_{\phi} \sim 1 TeV$ imposes the bound $\eta
\lsim 3 \times 10^{11} GeV$. Similar constraints are obtained for global
topological defects. Some non-standard cosmological scenarios which can avoid
these constraints are pointed out.
