  We study the effects of random bonds on spin chains that have an excitation
gap in the absence of randomness. The dimerized spin-1/2 chain is our principal
example. Using an asymptotically exact real space decimation renormalization
group procedure, we find that dimerization is a relevant perturbation at the
random singlet fixed point. For weak dimerization, the dimerized chain is in a
Griffiths phase with short range spin-spin correlations and a divergent
susceptibility. The string topological order, however, is not destroyed by bond
randomness and dimerization is stabilized by the confinement of topological
defects. We conjecture that random integer spin chains in the Haldane phase
exhibit similar thermodynamic and topological properties.
